Diagonal dominance

Lie theory studies continuous symmetries through Lie groups, Lie algebras, and their representations. A rotation can change by an arbitrarily small angle: its parameter varies continuously, yet composing two rotations obeys an exact group operation. The central question is how such finite transformations are controlled by their velocities at the identity element.

An infinite group need not require infinitely many independent parameters. The planar rotation group has infinitely many elements but only one local angle coordinate. More generally, an

n

-dimensional Lie group has an

n

-dimensional tangent space

g = T_{e} G

at its identity element. A basis of this Lie algebra, together with the Lie brackets between its basis vectors, supplies finite data for infinitesimal composition. These brackets record the first correction to adding small transformation parameters. The Exponential map of a Lie group turns a chosen infinitesimal direction into an actual motion.

Finite tangent data consists of a finite basis and a finite table of Lie brackets. For planar rotation, every generator is

a J

with

a \in R

, using the quarter-turn matrix

J

computed below: one real coefficient specifies any angular velocity. This Lie algebra has infinitely many vectors, but its dimension is one. Linear combinations of the chosen basis vectors supply all the possible infinitesimal motions.

This passage between geometry and algebra is useful because linear maps and matrix commutators are easier to compute than an entire family of transformations. The local information does not determine every global identification: the additive real line and the circle group have isomorphic one-dimensional Lie algebras, but only the latter wraps around. For the local-versus-global viewpoint and the connection with differential equations, see Meinrenken, Lie groups and Lie algebras, introduction and Section 4.

Linear algebraic group

An affine algebraic group is an affine variety

G

whose multiplication and inversion are regular maps. Equivalently, its coordinate ring

k [G]

is a commutative Hopf algebra, with comultiplication induced by multiplication in

G

Adjoint Chevalley group

Fix a Chevalley basis and its integral lattice. The adjoint Chevalley group over a field is generated by the root operators obtained from the integral divided-power polynomials

x_{α} (t) = \sum_{j} t^{j} (ad e_{α})^{j} / j!

on that lattice. The coefficients are first computed over the integers and then reduced to the field; factorials need not be invertible there. This is the adjoint construction; other weight lattices give other isogeny forms of groups of Lie type.

Adjoint Chevalley group of type A1 ( $A_{1} (K) ≅ S L_{2} (K) / {a I : a^{2} = 1}$ )

On the sl2 Lie algebra with basis

e = E_{12}, h = E_{11} - E_{22}, f = E_{21}

, the integral adjoint root polynomials are

x_{+} (t) e x_{-} (s) f = e, = f, x_{+} (t) h x_{-} (s) h = h - 2 t e, = h + 2 s f, x_{+} (t) f x_{-} (s) e = f + t h - t^{2} e, = e - s h - s^{2} f .

These are conjugation by the upper and lower elementary matrices of the special linear group. Those matrices generate

S L_{2} (K)

, and a matrix commuting with both

E_{12}

and

E_{21}

is scalar. Thus the kernel is precisely

{a I : a^{2} = 1}

over every field, even in characteristic two, and the image is the displayed quotient.

Algebraic one-parameter subgroup ( $λ : G_{m} \to G$ )

An algebraic one-parameter subgroup is a regular group homomorphism from the multiplicative algebraic group. In

S L_{n}

, a change of basis makes it

t \mapsto diag (t^{a_{1}}, \dots, t^{a_{n}})

for integers with sum zero. This is the algebraic multiplicative notion, distinct from an additive real-parameter subgroup of a Lie group.

Geometrically reductive algebraic group

For every finite-dimensional rational representation

V

and every nonzero

v \in V^{G}

, a geometrically reductive algebraic group admits a homogeneous invariant polynomial

F \in k [V]^{G}

of positive degree with

F (v) \neq = 0

. A linearly reductive group can take degree one. In positive characteristic, higher degree is essential: a power

p^{r}

can separate a fixed vector even when an invariant linear functional cannot.

Determinant separation for positive-characteristic SL2 ( $q = p^{r}, det : End (Sym^{q - 1} k^{2}) \to k$ )

Choose a torus-invariant functional nonzero on a fixed vector, and map the representation to right-torus-invariant matrix coefficients on

S L_{2}

. These functions lie in a finite filtration by polynomials of bidegree

(n, n)

in the two matrix columns. For

n = q - 1

, all binomial coefficients are nonzero in characteristic

p

, so the invariant tensor given by the determinant to power

n

induces an invertible map from the symmetric power's dual to itself. The filtration piece becomes its endomorphism space, with the constant function corresponding to the identity. The endomorphism determinant is an invariant degree-

q

polynomial equal to one there; composing it with the original representation map proves geometric reductivity.

Linearly reductive algebraic group

An affine algebraic group is linearly reductive when every finite-dimensional rational representation is completely reducible. Equivalently its invariants functor is exact. Over a field of characteristic zero, reductive algebraic groups are linearly reductive; in positive characteristic this stronger property fails for many reductive groups. An algebraic torus is linearly reductive in every characteristic because its rational representations decompose into character spaces.

Rational structure on an algebraic group

A rational structure over

F_{q}

is a model over that finite field together with its identification after extension to

\overline{F}_{q}

. It determines a Frobenius endomorphism of an algebraic group, whose fixed points are the rational points. Conjugating a standard Frobenius by an algebraic group automorphism transports the rational structure, even if the automorphism is not itself rational for the original structure.

Frobenius endomorphism of an algebraic group ( $F : G \to G$ )

For an algebraic group defined over

F_{q}

, its Frobenius morphism acts as entrywise

q

th powers in a standard matrix realization. Its differential is zero, and its fixed-point group is finite. A conjugate

F^{'} = φF φ^{- 1}

defines an isomorphic rational structure. This group morphism must be distinguished from a field Frobenius endomorphism alone and from the convention called geometric Frobenius in Galois theory, which is inverse to the arithmetic generator on field points.

Lang theorem for algebraic groups

For a connected smooth algebraic group and a finite-field Frobenius

F

, the Lang map

g \mapsto g^{- 1} F (g)

is surjective. One proof uses the twisted action

x \cdot y = x y F (x)^{- 1}

. Since

d F = 0

, every orbit map has surjective differential, so each orbit is open; connectedness leaves only one orbit. The same differential calculation makes the Lang map étale, with finite fibres equal to left cosets of

G^{F}

Lang map ( $L_{F} (g) = g^{- 1} F (g)$ )

For the Lang map,

L_{F} (g h) = h^{- 1} L_{F} (g) F (h)

. This twisted right-translation identity determines which subgroups act on its inverse images. Fibres are left

G^{F}

-cosets, by the Lang theorem for algebraic groups.

Affine group ( $Aff (V) = V ⋊ GL (V)$ )

For a finite-dimensional vector space

V

, the affine group consists of invertible affine maps

x \mapsto A x + b

, where

A

belongs to the general linear group and

b \in V

. Its product is

(A, b) (A^{'}, b^{'}) = (A A^{'}, b + A b^{'})

, making it the semidirect product of the translation vector space and its general linear group. On the real line the positive-dilation subgroup is the real affine group used in the adjacent article.

Rational representation ( $ρ : G \to G L (W)$ )

A rational representation is a group homomorphism whose matrix coefficients are regular functions on the affine algebraic group. For the general linear group, those functions are polynomials in matrix entries and the inverse determinant. The word rational does not permit poles at points of the group.

Unstable vector in a rational representation ( $0 \in \overline{G v}$ )

A vector is unstable if zero lies in the Zariski closure of its orbit. For complex

S L_{n}

, this is equivalent to an algebraic one-parameter subgroup taking it to zero as

t \to 0

. In the subgroup's weight decomposition, exactly the positive-weight components can occur in such a vector.

Hilbert-Mumford criterion for the affine null cone

For complex

S L_{n}

, choose a sequence taking an unstable vector to zero and apply singular value decomposition to each group element. Compactness gives a limiting right-unitary factor. Every nonzero torus-weight component of the transformed vector forces a strict linear inequality on the diagonal exponents. A rational solution of these finitely many strict inequalities scales to integral exponents with sum zero, producing an algebraic one-parameter subgroup that takes the vector to zero. The converse follows immediately from the subgroup limit.

Root-multiplicity criterion for unstable binary forms ( $f unstable ⟺ f = 0 or a root has multiplicity > d /2$ )

For

diag (t, t^{- 1})

, the monomial

x^{d - i} y^{i}

has weight

d - 2 i

. Positive weights force divisibility by

x^{⌊ d /2 ⌋ + 1}

. Conjugating the subgroup replaces

x

by any nonzero linear factor. Thus a nonzero binary form is unstable exactly when a projective root has multiplicity greater than half its degree. Equality gives zero weight and does not imply instability.

Complete reducibility of rational GL and SL representations

For finite-dimensional rational representations of complex

G L_{m}

S L_{m}

, average a Hermitian inner product over the compact group

U (m)

S U (m)

using normalized Haar measure. Orthogonal complements become invariant under the compact group and its complexified Lie algebra. The latter generates the complex group, giving invariant complements and hence complete reducibility. This conclusion does not hold for arbitrary affine algebraic groups, such as the additive group.

Rational representation of the general linear group ( $ρ_{ij} \in C [g_{ab}, det (g)^{- 1}]$ )

The matrix coefficients of a rational

G L_{m}

action belong to its coordinate ring

C [g_{ab}, det (g)^{- 1}]

. Multiplying the action by a sufficiently large determinant twist clears all determinant denominators and produces a polynomial representation. Irreducible rational actions are indexed by decreasing integer highest weights, with negative coordinates allowed.

Rational Schur module ( $D_{λ} = det^{- k} \otimes D_{λ + k 1}$ )

For a dominant integer tuple

λ

, choose

k = max (0, - λ_{m})

and define

D_{λ} = det^{- k} \otimes D_{λ + k (1, \dots, 1)}

, using the polynomial Schur module on the right. The alternant character formula for the general linear group shows that a larger admissible shift yields the same irreducible representation.

Dual of a rational Schur module ( $D_{λ}^{*} ≅ D_{(- λ_{m}, \dots, - λ_{1})}$ )

The dual of

D_{λ}

has dominant label

- w_{0} λ = (- λ_{m}, \dots, - λ_{1})

. Substitution

x_{i} \mapsto x_{i}^{- 1}

in the alternant character formula for the general linear group and reversal of determinant columns gives that character. The numerator and denominator reversal signs and common monomial factors cancel.

Highest-weight classification of rational GL representations ( $λ \in Z^{m}, λ_{1} \geq \dots \geq λ_{m}$ )

The irreducible rational

G L_{m}

modules are

D_{λ} (V) = (det V)^{λ_{m}} \otimes D_{λ - λ_{m} (1, \dots, 1)} (V)

. The second factor is a Schur module. Clearing determinant denominators reduces completeness to the Schur algebra classification of polynomial modules. Distinct dominant integer tuples have distinct highest torus weights. Their characters are symmetric Laurent polynomials; only polynomial modules have characters defined at every singular endomorphism.

Rational extension from SL to GL ( $ρ^{'} (g) ∣_{W_{k}} = a^{k} ρ (a^{- 1} g) ∣_{W_{k}}, a^{m} = det g$ )

Split a rational

S L_{m}

representation by its finite scalar center: on

W_{k}

ρ (ζ I) = ζ^{k} I

for

ζ^{m} = 1

. The displayed extension is independent of the scalar-root choice and is a group homomorphism. A matrix coefficient of

ρ ∣_{W_{k}}

can be represented on

S L_{m}

by a sum of homogeneous polynomials

P_{d}

with

d \equiv k (mod m)

, by averaging over the scalar center. Its extension is

\sum_{d} (det g)^{(k - d) / m} P_{d} (g)

, a regular function on

G L_{m}

. Every invariant subspace remains invariant under this extension, so irreducibility is preserved in both directions.

Determinant twist ( $g ⟼ (det g)^{r} ρ (g)$ )

Tensoring a rational representation of the general linear group with a determinant power shifts every highest-weight coordinate by the same integer

r

. Positive twists can clear determinant denominators; negative twists supply rational representations that do not extend to singular matrices. Twisting preserves irreducibility and dimension.

One-dimensional rational characters of the general linear group ( $X^{*} (G L_{m}) = Z \cdot det$ )

Every one-dimensional rational

G L_{m}

representation is

det^{r}

for one integer

r

. Its restriction to the diagonal torus is a Laurent monomial. Invariance under conjugation by permutation matrices forces all its exponents to agree, and density of diagonalizable invertible matrices gives the result on the whole group. For

G L_{1}

, the character identity

p (z w) = p (z) p (w)

directly forces a Laurent polynomial to be one monomial with coefficient one.

Polynomial representation of the general linear group ( $ρ_{ij} \in C [g_{ab}]$ )

In a polynomial representation, every matrix coefficient is a polynomial in the matrix entries. The action extends to the monoid of all matrices. The scalar action splits it into homogeneous polynomial degrees; each homogeneous degree-

n

category is the module category of the Schur algebra

S (m, n)

. Its irreducibles are the Schur modules

D_{λ} (C^{m})

with

λ ⊢ n

and at most

m

rows.

Binary form ( $f (x, y) = \sum_{i = 0}^{d} (i d) f_{i} x^{d - i} y^{i}$ )

A binary form is a homogeneous polynomial in two variables. Degree-

d

binary forms give the symmetric-power representation of the standard two-dimensional representation of

S L_{2}

, after using its invariant alternating form to identify the standard representation with its dual.

Binary quartic ( $f (X, Y) = \sum_{j = 0}^{4} a_{j} X^{4 - j} Y^{j}$ )

A binary quartic is a homogeneous degree-four polynomial in two variables. Over the complex numbers it determines four points of the projective line, counted with multiplicity. Its discriminant is nonzero exactly when the four points are distinct; this is also its geometric invariant theory stability condition under

S L_{2}

Multiplicity-free symmetric-algebra model for polynomial GL representations ( $Sym (V \oplus Λ^{2} V) = ⨁_{λ} D_{λ} (V)$ )

The symmetric algebra on

V \oplus Λ^{2} V

contains every irreducible polynomial representation of

G L (V)

exactly once. Its torus character is

\prod_{i} (1 - x_{i})^{- 1} \prod_{i < j} (1 - x_{i} x_{j})^{- 1}

, which equals

\sum_{λ} s_{λ} (x)

by the Schur identity. Complete reducibility and linear independence of Schur characters identify the summands. This is a formal graded identity: each fixed scalar degree is finite, so no analytic convergence is needed. Multiplicity one refers to irreducible modules, not arbitrary reducible representations.

Schur functor ( $D_{λ} : V ⟼ h_{t} V^{\otimes n}$ )

For a partition of an integer

λ ⊢ n

, applying a Young symmetrizer of that shape to the tensor power of each vector space gives a Schur functor. The construction respects linear maps because their tensor powers commute with place permutations. For single rows it gives symmetric powers, and for single columns exterior powers.

Schur module ( $D_{λ} (V)$ )

A Schur module is the value of a Schur functor on a vector space. Over the complex numbers it is either zero or an irreducible polynomial representation of the general linear group. Its character is the Schur polynomial

s_{λ}

in the eigenvalues, and its highest weight is

λ

padded with zeros.

Length bound for a Schur module ( $D_{λ} (C^{m}) \neq = 0 ⟺ ℓ (λ) \leq m$ )

A column of length greater than

m

antisymmetrizes more than

m

vectors, giving zero. When there are at most

m

rows, place the

i

th basis vector in every tensor position of row

i

. Row symmetrization multiplies by a nonzero factorial product, and column antisymmetrization is nonzero because the vectors in every column are distinct. This proves the precise nonvanishing criterion for a Schur module.

Schur algebra ( $S (m, n) = End_{C S_{n}} ((C^{m})^{\otimes n})$ )

The Schur algebra is the commutant of place permutations on a tensor power. It is the linear span of the diagonal general linear group action, by Schur–Weyl duality. Modules over

S (m, n)

correspond to homogeneous degree-

n

polynomial representations. Over the complex numbers it is a semisimple algebra: decompose the tensor power as a module for the semisimple group algebra

C S_{n}

, and take its endomorphism algebra.

Invertible tensor powers span the Schur algebra ( $S (m, n) = span {g^{\otimes n} : g \in GL (V)}$ )

The commutant of the permutation action on

V^{\otimes n}

identifies with the symmetric tensors in

End (V)^{\otimes n}

. Polarization spanning of symmetric tensors spans it by

a^{\otimes n}

. Interpolating

(a + t I)^{\otimes n}

n + 1

values where

a + t I

is invertible expresses each such power using invertible ones. Hence the Schur algebra is the linear span of the diagonal general linear action.

Schur–Weyl duality ( $V^{\otimes n} = ⨁_{λ ⊢ n} S^{λ} \otimes D_{λ} (V)$ )

The commuting actions of the symmetric group and the general linear group on a tensor power are mutual commutants. The displayed sum ranges over partitions with at most

dim V

rows. The Specht modules and Schur modules are the simple factors for the two actions. In particular a primitive Young symmetrizer selects one copy of the matching Schur module.

Tensor cube of the defining unitary representation ( $V^{\otimes 3} = Sym^{3} V \oplus Λ^{3} V \oplus V_{(2, 1)}^{\oplus 2}$ )

For the unitary group

U (n)

with

n \geq 3

, the third tensor power of its defining representation decomposes as displayed. The first two summands have highest weights

(3, 0, \dots)

and

(1, 1, 1, 0, \dots)

and dimensions

(3 n + 2)

and

(3 n)

. The mixed summand has highest weight

(2, 1, 0, \dots)

and dimension

n (n^{2} - 1) /3

. In the three-dimensional weight space spanned by permutations of

e_{1} \otimes e_{1} \otimes e_{2}

, the vectors killed by all positive root vectors have coefficient sum zero, giving multiplicity two. For

n = 2

, the exterior cube vanishes and

V_{(2, 1)} = det \otimes C^{2}

Trace of a permuted tensor power ( $tr (σ ξ^{\otimes n}) = \prod_{r} tr (ξ^{r})^{α_{r}}$ )

σ

has

α_{r}

cycles of length

r

, contraction of matrix entries around its cycles gives

tr (σ ξ^{\otimes n}) = \prod_{r} tr (ξ^{r})^{α_{r}}

. The formula holds for every endomorphism, not only diagonalizable ones. The Schur–Weyl duality decomposition equates it with a sum of products of Specht and Schur characters.

Young-symmetrizer tensor decomposition ( $V^{\otimes n} = ⨁_{λ, t standard} h_{t} V^{\otimes n}$ )

The standard-tableau right-ideal decomposition of

C S_{n}

, tensored over that algebra with

V^{\otimes n}

, gives this direct sum of general linear group modules. With

e_{t} = h_{t} / H_{λ}

, the map

e_{t} C S_{n} \otimes T \to e_{t} T

is an isomorphism. Individual summands generally need not be symmetric group submodules; instead each is a Schur module for the commuting action.

Lie algebra of an affine algebraic group

The Lie algebra of an affine algebraic group

G

is the tangent space

T_{e} G

at its identity. Its bracket is the commutator of the corresponding left-invariant derivations of the coordinate ring.

Descent of an irreducible SL2 representation to PGL2

The irreducible rational representations of

S L_{2} (C)

are

Sym^{n} (C^{2})

. Such a representation descends through

S L_{2} \to PG L_{2}

exactly when the central element

- I

acts trivially, which holds exactly when

n

is even.

Faithful representation of an affine algebraic group

Every affine algebraic group has a faithful finite-dimensional rational group representation. Choose algebra generators of

k [G]

and place them in a finite-dimensional subspace stable under right translations. An element acting trivially on that subspace has the same coordinate values as the identity and therefore is the identity.

Left-right regular representation of an affine algebraic group

The group

G \times G

acts on

k [G]

((a, b) f) (x) = f (a^{- 1} x b) .

For

G = G_{m}

, this gives

k [G] = ⨁_{r \in Z} k_{(- r, r)}

. For

G = G_{a}

, the action factors through

(a, b) \mapsto b - a

and the degree filtration of

k [t]

has trivial one-dimensional successive quotients.

Semisimple element of an affine algebraic group

An element of an affine algebraic group is semisimple when its image in one, equivalently every, faithful finite-dimensional representation is a diagonalizable linear map.

Unipotent element of an affine algebraic group

An element of an affine algebraic group is unipotent when its image in one, equivalently every, faithful finite-dimensional representation has every eigenvalue equal to one.

Unipotent matrix ( $(U - I)^{d} = 0$ )

A square matrix

U

is unipotent when

U - I

is nilpotent. Over an algebraically closed field, this is equivalent to all eigenvalues being one. If

j

belongs to a nilpotent ideal of an operator algebra,

I + j

is a unipotent matrix. This realizes the radical factor in the Levi decomposition of a quiver automorphism group.

Jordan decomposition in an affine algebraic group

This is the multiplicative Jordan–Chevalley decomposition in a linear algebraic group. Every element

g

of an affine algebraic group has unique commuting semisimple and unipotent parts

g_{s}, g_{u}

such that

g = g_{s} g_{u}

. The construction agrees with the multiplicative Jordan decomposition in every rational representation.

Derived subgroup of an affine algebraic group

The derived subgroup

[G, G]

is the closed algebraic subgroup generated by the commutators

x y x^{- 1} y^{- 1}

. If

G

is connected, images of finite products of commutators are connected and their closures stabilize by dimension, which shows that

[G, G]

is connected.

Solvable affine algebraic group

An affine algebraic group is solvable when its derived series reaches the identity. A connected solvable affine algebraic group can be conjugated into an upper triangular matrix group by the Lie-Kolchin theorem.

Lie-Kolchin theorem

Every connected solvable affine algebraic subgroup of

GL (V)

preserves a complete flag in

V

, equivalently it is conjugate to a subgroup of the upper triangular matrices.

Diagonalizable group

A diagonalizable algebraic group is isomorphic to a closed subgroup of a product of copies of

G_{m}

. Every rational representation of such a group decomposes into weight spaces.

Reductive group

A reductive algebraic group is a smooth connected affine algebraic group whose largest connected normal unipotent algebraic group is trivial.

Finite group of Lie type ( $G^{F}$ )

A finite group of Lie type is obtained from rational points of a connected reductive algebraic group with a suitable Frobenius endomorphism of an algebraic group, often followed by central quotient or passage to a derived subgroup. Its complex representations combine finite-group character theory with the geometry of Borel subgroups, maximal algebraic tori, and the Weyl group.

Harish-Chandra induction ( $R_{L \subset P}^{G}$ )

For a rational parabolic subgroup

P = L ⋉ U

, use inflation of a group representation for a complex

L^{F}

-module across

P^{F} \to L^{F}

and induce it to

G^{F}

. This is Harish-Chandra induction. Its adjoint, Harish-Chandra restriction, takes

U^{F}

-invariants. These are ordinary finite-group group representation functors and should not be confused with general Deligne-Lusztig induction from a nonrational Borel.

Mackey formula for Harish-Chandra induction

For rational parabolic subgroups

P = LU

and

Q = M V

, define

S (M, L) = {x : M \cap^{x} L contains a maximal algebraic torus of G}

. Then

^{*} R_{M}^{G} R_{L}^{G}

is the sum over

M^{F} \ S (M, L)^{F} / L^{F}

R_{M \cap^{x} L}^{M}^{*} R_{M \cap^{x} L}^{^{x} L} ad x

, with the intersection parabolics understood. It follows from ordinary finite-group Mackey restriction formula, taking

V^{F}

-invariants, and the compatible-Levi intersection lemma for parabolics.

Harish-Chandra restriction ( $^{*} R_{L \subset P}^{G} (V) = V^{U^{F}}$ )

Restriction first views a

G^{F}

-module as a

P^{F}

-module, then takes the subspace fixed by the unipotent radical. The resulting

L^{F}

-module is adjoint to Harish-Chandra induction by ordinary Frobenius reciprocity. Over characteristic zero, averaging makes invariants exact and identifies them with coinvariants.

Cuspidal representation of a finite reductive group

An irreducible complex

G^{F}

-module is cuspidal if its Harish-Chandra restriction is zero for every proper rational parabolic subgroup. By adjunction, it occurs in no Harish-Chandra induction from a proper rational Levi subgroup. This is a finite-group notion, distinct from a cuspidal automorphic representation.

Deligne-Lusztig theory

Deligne-Lusztig theory constructs virtual characters of finite groups of Lie type from compactly supported étale cohomology of varieties associated with Frobenius and flags. Rational maximal tori and their finite-group characters provide induction data, while the geometry explains character orthogonality and cuspidality.

Deligne-Lusztig induction ( $R_{T}^{G} (θ)$ )

For a rational maximal torus

T

, choose a Borel subgroup containing it with unipotent radical

U

. The alternating compactly supported étale cohomology of

L_{F}^{- 1} (U)

, in the appropriate

T^{F}

-isotypic component, defines the virtual character of

G^{F}

R_{T}^{G} (θ)

. It is independent of the chosen Borel. When the Borel subgroup is rational, this is ordinary induction from its rational points after inflation of a group representation from

T^{F}

Harish-Chandra restriction of a Deligne-Lusztig character

For a rational Levi subgroup

M

of a rational parabolic subgroup and a rational maximal torus

T

, the torus case of the induction-restriction formula is

^{*} R_{M}^{G} R_{T}^{G} (θ) = \sum_{x \in M^{F} \ {g \in G^{F} :^{g} T \subset M} / T^{F}} R_{^{x} T}^{M} (^{x} θ) .

In particular it vanishes if there is no rational conjugate of

T

M

. If

T \subset M

, transitivity gives

R_{T}^{G} = R_{M}^{G} R_{T}^{M}

. Together with adjunction these identities show that an irreducible character up to sign arising from

T

is cuspidal exactly when

T

is elliptic.

Principal series of a finite reductive group

For a rational Borel subgroup

B = T U

, the principal series is

Ind_{B^{F}}^{G^{F}} θ

, with

θ

inflated from a linear character of

T^{F}

. The Mackey restriction formula and rational Bruhat decomposition give its self-inner-product as the size of the Weyl stabilizer of

θ

. It is irreducible precisely when that stabilizer is trivial.

Irreducibility of the split SL3 principal series

For

N = q - 1

, write a diagonal-torus character as

θ (diag (a, b, (ab)^{- 1})) = χ (a)^{r} χ (b)^{s}

, with

χ

a generator of the character group of

F_{q}^{\times}

. Its Weyl stabilizer is trivial exactly when

r, s, r - s

are all nonzero modulo

N

and, if

3 ∣ N

, the unordered pair

{r, s}

is not

{N /3, 2 N /3}

. The first conditions exclude transpositions; the exceptional pair excludes cyclic permutation of three characters modulo a common twist.

Deligne-Lusztig variety ( $X (w) = {B^{'} : (B^{'}, F (B^{'})) \in O (w)}$ )

For a connected reductive algebraic group and a Weyl group element

w

, this locally closed subset of the flag variety consists of flags in relative position

w

to their Frobenius image. For rational

B

it is

L_{F}^{- 1} (B \overset{w}{˙} B) / B

, using the twisted right action identity of the Lang map. Its dimension is the length of

w

Unipotent quotient in a Deligne-Lusztig variety

Choose rational

B = T U

, a representative

\overset{w}{˙}

, and

x

with

x^{- 1} F (x) = \overset{w}{˙}

. Set

T^{'} = x T x^{- 1}

and

U^{'} = F (xU x^{- 1})

. Then

T^{'}

is rational, and

L_{F}^{- 1} (U^{'}) / [(U^{'} \cap F^{- 1} (U^{'})) T^{' F}] ≅ X (w) .

The map sends

h

h x B

. Quotienting only by the finite group

T^{' F}

leaves fibres isomorphic to the positive-dimensional unipotent intersection when that intersection is nontrivial. For

w = 1

SL_{2}

, this is an affine-line fibre over each rational flag, so the missing quotient cannot be ignored in a variety isomorphism.

G

and a Borel subgroup

B

, the quotient

G / B

is its complete flag variety. Quotients

G / P

by parabolic subgroups are partial flag varieties.

Root datum

A root datum is a quadruple

(X^{*}, Φ, X_{*}, Φ^{\lor})

of dual lattices, roots, and coroots with the natural perfect pairing and reflection axioms. A reductive group with maximal torus

T

has

X^{*} = X^{*} (T)

and

X_{*} = X_{*} (T)

Bruhat decomposition

For a connected reductive algebraic group, a Borel subgroup

B

, and a maximal torus

T \subseteq B

, the Weyl group

W = N_{G} (T) / T

indexes the double cosets:

G = ⨆_{w \in W} B \overset{w}{˙} B .

Relative position of Borel subgroups

The relative position of Borel subgroups

B_{1}, B_{2}

is the Weyl group index of their simultaneous conjugacy orbit. After moving the first subgroup to a fixed

B

, the second has the form

g B g^{- 1}

and its index is the double coset

B g B

. This identifies the diagonal action on pairs of flags with the Bruhat decomposition.

Bruhat decomposition of a BN-pair

This is the Bruhat decomposition associated with a BN-pair. The Bruhat decomposition is the disjoint union

G = ⨆_{w \in W} B \overset{w}{˙} B

associated with a BN-pair.

Algebraic-group torsor

For an algebraic group

H

, a flat

H

-torsor over

Y

is a faithfully flat morphism

X \to Y

with a right

H

-action such that

X \times H \to X \times_{Y} X

(x, h) \mapsto (x, x h)

, is an isomorphism. A Zariski torsor additionally becomes

U \times H

on a Zariski open cover of

Y

Unipotent

An element

u

of a ring is unipotent when

u - 1

is nilpotent. For a matrix, this says that every eigenvalue is one. A unipotent algebraic group consists of unipotent elements in a faithful linear representation.

Unipotent algebraic group

A unipotent algebraic group is built from unipotent elements. A unipotent algebraic group is an affine algebraic group all of whose elements are unipotent. The Kolchin theorem embeds it into an upper unitriangular group.

Unipotent radical ( $R_{u} (G)$ )

It is the largest connected normal unipotent subgroup of a linear algebraic group. A connected reductive algebraic group has trivial unipotent radical. The subgroup

1 + J (E)

of a finite-dimensional algebra's unit group is closed, connected and unipotent.

Kolchin theorem

Every unipotent algebraic subgroup of

GL_{n}

is conjugate to a subgroup of the upper unitriangular group. The filtration by vanishing initial superdiagonals then proves that every unipotent algebraic group is a nilpotent group.

Lie group

 2  0

A Lie group is a group that is also a finite-dimensional smooth manifold, with multiplication

G \times G \to G

and inversion

G \to G

both smooth maps. Its tangent space at the identity element carries a natural Lie algebra structure. Smoothness lets us differentiate the group operation; being a group means that every transformation can be composed and undone. The dimension of a manifold counts independent local parameters, rather than the number of elements of the group. See MIT 18.745, Lecture 2, Definition 2.1 and Example 2.5.

For example, the planar special orthogonal group

SO (2)

consists of rotation matrices

R_{θ} = (cos θ sin θ - sin θ cos θ), R_{α} R_{β} = R_{α + β}, R_{θ}^{- 1} = R_{- θ} .

The sine and cosine entries vary smoothly. Four entries appear, but they are constrained by

R_{θ}^{T} R_{θ} = I

and

det R_{θ} = 1

: locally, one angle specifies the whole matrix. Multiplication adds angles, with angles differing by

2 π

representing the same rotation. Thus

SO (2)

is an infinite group of dimension one, geometrically a circle group.

The positive-dimensional examples motivate the language of continuous symmetry, but the definition also admits finite groups with the discrete topology as zero-dimensional Lie groups. Conversely, being an infinite group alone says nothing about having a continuous angle parameter. The compatibility between smooth maps and the group operation is what makes differentiation useful.

Lie groups have no small subgroups

Choose an injective local exponential chart on a ball of radius

r

in the Lie algebra, and let

0 < ε < r /2

. The neighborhood

exp (B_{ε})

contains no nontrivial subgroup. Otherwise it contains an element

exp X

with

0 < ∥ X ∥ < ε

. Choose the first integer

m

with

m ∥ X ∥ \geq ε

. Then

ε \leq m ∥ X ∥ < 2 ε < r

, so chart injectivity puts

(exp X)^{m} = exp (m X)

outside

exp (B_{ε})

, a contradiction. Zero-dimensional Lie groups have the singleton identity as such an open neighborhood.

G2 (mathematics, $G_{2}$ )

G_{2}

denotes an exceptional Lie group and its associated Lie algebra, of rank two and dimension fourteen. Its roots form the G2 root system. The compact real form is the automorphism group of the octonion algebra; other real forms and the complex Lie algebra have the same complex root-system type.

Lie group–Lie algebra correspondence ( $G ⟼ T_{e} G, ϕ ⟼ d_{e} ϕ$ )

 2  0

Differentiation sends Lie groups to Lie algebras and smooth group homomorphisms to bracket-preserving linear maps. On connected simply connected Lie groups, this is an equivalence with finite-dimensional real Lie algebras: the Lie third theorem gives existence, and integration of a Lie algebra homomorphism gives unique maps. General connected groups require discrete central quotient data, and disconnected groups additionally require component and conjugation data.

Lie third theorem ( $Lie (H) ≅ h$ )

Every finite-dimensional real Lie algebra is the Lie algebra of a connected simply connected Lie group. One construction first uses the Ado theorem to embed it into a matrix Lie algebra, integrates the corresponding left-invariant distribution by the Frobenius theorem to a connected immersed Lie subgroup, and then takes a universal covering Lie group. The immersed subgroup need not be closed; taking its ambient closure may change its Lie algebra.

Universal covering Lie group ( $p : G ⟶ G^{\circ}$ )

The universal cover of a connected Lie group carries a unique lifted Lie group structure with a chosen identity, making its covering map a Lie group homomorphism. The kernel is a discrete central subgroup: conjugation of any kernel element is a continuous map from a connected group into a discrete set, hence constant. Every connected Lie group is therefore a quotient of its simply connected cover by a discrete central subgroup.

Complex Lie group

A complex Lie group is a complex manifold whose multiplication and inversion are holomorphic maps between complex manifolds. Its tangent space at the identity is a complex Lie algebra; the Adjoint representation of a Lie group and the Lie exponential map are holomorphic. Compatibility with a complex manifold structure is essential: an arbitrary even-dimensional real Lie group need not have holomorphic multiplication.

Compact connected complex Lie groups are complex tori ( $G ≅ C^{n} / B, B ≅ Z^{2 n}$ )

Every scalar holomorphic function on a compact connected complex manifold is constant by the maximum modulus principle. Apply this to each entry of the holomorphic Adjoint representation of a Lie group; its value is constantly the identity. Differentiation makes the complex Lie algebra abelian, hence connectedness makes the group abelian. The holomorphic Lie exponential map identifies it with

C^{n}

modulo a discrete subgroup. Compactness forces that subgroup to span the underlying real vector space, giving a full-rank Euclidean lattice and a complex torus.

Abelian Lie group

An abelian Lie group has commutative multiplication. Its Lie algebra is an abelian Lie algebra, and its Lie exponential map is a group homomorphism from the additive Lie algebra. Conversely, an additive Lie exponential map forces the identity component of a Lie group to be abelian, but does not force all disconnected components to commute.

Classification of connected abelian Lie groups ( $G ≅ R^{n} /Λ ≅ T^{r} \times R^{n - r}$ )

For a connected abelian Lie group, the Lie exponential map is an additive group homomorphism and a local diffeomorphism at zero. Its image is an open subgroup, hence the whole connected group. Its kernel is a discrete additive subgroup of a real vector space. A basis of that kernel identifies it with

Z^{r}

inside

R^{n}

, giving the displayed classification. The compact factors record periods of one-parameter subgroups.

Sol Lie group ( $R^{2} ⋊ R$ )

The multiplication

(x^{'}, y^{'}, z^{'}) (x, y, z) = (x^{'} + e^{- z^{'}} x, y^{'} + e^{z^{'}} y, z^{'} + z)

defines the three-dimensional Sol Lie group. Its left-invariant vector fields

e^{- z} \partial_{x}, e^{z} \partial_{y}, \partial_{z}

have dual coframe

e^{z} d x, e^{- z} d y, d z

. Their sum of squares is a left-invariant metric,

d s^{2} = e^{2 z} d x^{2} + e^{- 2 z} d y^{2} + d z^{2}

Semisimple Lie group

A real or complex Lie group is semisimple when its Lie algebra is semisimple. For a real group, the Killing form then defines a nondegenerate bi-invariant pseudo-Riemannian metric. Compact semisimple groups have negative-definite Killing form; noncompact examples such as

S L (2, R)

have indefinite Killing form. A disconnected group can also have a semisimple Lie algebra, so connectedness is a separate hypothesis when integrating infinitesimal actions.

Free step-N nilpotent Lie group ( $G^{N} (V) = exp (g^{N} (V))$ )

Inside the truncated tensor algebra, let

g^{N} (V)

be the span of iterated Lie brackets of elements of

V

of lengths at most

N

. Its exponential is the connected simply connected free nilpotent Lie group of step

N

, with tensor multiplication. The bracket is the commutator in the tensor algebra. Its grading defines dilations that multiply degree

k

λ^{k}

Step-2 planar group coordinates ( $(v, a) (w, b) = (v + w, a + b + \frac{1}{2} (v_{1} w_{2} - v_{2} w_{1}))$ )

Write a group element as

exp (v + a [e_{1}, e_{2}])

. The step-two Baker-Campbell-Hausdorff formula gives the displayed multiplication and inverse

(- v, - a)

. The scalar

a

is the antisymmetric second signature coordinate, which equals the signed area for a closed planar path. The Carnot-Carathéodory distance is comparable to

∣ v ∣ + ∣ a ∣^{1/2}

near the identity, with the same homogeneous scaling globally.

Loop group ( $L G = C^{\infty} (S^{1}, G)$ )

The loop group of a Lie group is the group of smooth maps from the circle to that group, with pointwise multiplication and the smooth-function topology. Rotation of the source circle acts by group automorphisms. For

G = U (1)

, every loop has the form

z^{w} e^{ih (z)}

, where

w \in Z

is its winding number and

h

is a smooth real periodic function determined up to addition of an integer multiple of

2 π

Positive energy representation of the loop group of U(1) ( $R_{t} π (g) R_{t}^{*} = π (g (e^{i t} \cdot))$ )

Such a continuous projective unitary representation admits strongly continuous rotations whose self-adjoint generator is bounded below and whose adjoint action rotates the loops. In the basic fermionic construction, integer one-particle modes are filled below zero. The rotation energy counts nonnegative particle energies and positive hole energies. It has finite-dimensional integer eigenspaces and implements circle rotations. Tensor products give higher nonnegative integer levels.

Loop group two-cocycle for U(1) ( $c ((w, h), (v, k)) = e^{i v h_{0} - iω (h, k) /2}$ )

Write a loop as

z^{w} e^{ih}

with

h

smooth real periodic and

h_{0}

its mean. For the level-one representation with modes filled below zero and

ω (h, k) = (2 π)^{- 1} \int_{0}^{2 π} h^{'} k

, choose implementers

U (w, h) = S^{w} e^{i A (h)}

, where

S

increases charge by one,

SQ S^{*} = Q - I

, and

[A (h), A (k)] = iω (h, k) I

. Their product differs from

U (w + v, h + k)

by the displayed two-cocycle. Its phase is bilinear, which verifies the cocycle identity; integer

v

makes it independent of the

2 π

ambiguity in

h_{0}

. Rephasing implementers changes it by a group coboundary.

Bi-invariant pseudo-Riemannian metric

A nondegenerate invariant symmetric bilinear form on a Lie algebra extends by translation to a metric preserved by both left and right translation. It may have indefinite signature. Its Levi-Civita connection is

\nabla_{X} Y = \frac{1}{2} [X, Y]

for left-invariant vector fields, and its Ricci tensor is

- K /4

. Choosing the nondegenerate Killing form of a semisimple Lie algebra yields a canonical indefinite Einstein manifold; compact groups instead admit the positive metric

- K

Neutral invariant metrics on the free two-step Lie algebra on three generators ( $signature (g) = (3, 3)$ )

Let

[e_{1}, e_{2}] = e_{6}

[e_{1}, e_{3}] = e_{4}

and

[e_{2}, e_{3}] = e_{5}

, with

e_{4}, e_{5}, e_{6}

central. The symmetric tensor

S = e_{1} \otimes e_{5} + e_{5} \otimes e_{1} + e_{3} \otimes e_{6} + e_{6} \otimes e_{3} - e_{2} \otimes e_{4} - e_{4} \otimes e_{2}

is invariant under the Adjoint representation of a Lie algebra: the derivation action of each of

e_{1}, e_{2}, e_{3}

cancels in pairs. For

t \neq = 0

Q = tS + \sum_{j = 4}^{6} a_{j} e_{j} \otimes e_{j}

has determinant

- t^{6}

and inverse metric

g = t^{- 1} S (λ) - t^{- 2} (a_{5} (λ^{1})^{2} + a_{4} (λ^{2})^{2} + a_{6} (λ^{3})^{2})

in the dual coframe. Its three hyperbolic planes give the displayed metric signature. On a connected Lie group this supplies a four-parameter family of bi-invariant pseudo-Riemannian metrics. Disconnected components must preserve the tensor separately; the automorphism reversing

e_{1}, e_{4}, e_{6}

while fixing the other generators reverses

S

and prevents this extension.

Killing-form Einstein metric ( $g = B, Ric = - \frac{1}{4} B$ )

On a real semisimple Lie group, the nondegenerate invariant Killing form gives a bi-invariant pseudo-Riemannian metric. For left-invariant fields the Koszul formula gives

\nabla_{X} Y = [X, Y] /2

and

R (X, Y) Z = - [[X, Y], Z] /4

. Taking the trace gives

Ric (Y, Z) = - B (Y, Z) /4

, proving the Einstein property. On a compact semisimple group,

- B

is positive definite and has Einstein constant

1/4

. The indefinite general construction is not a proof of existence of a positive-definite bi-invariant metric on a noncompact group.

Compact Lie group

A compact Lie group is a Lie group whose underlying manifold is compact. Averaging any positive inner product over its Haar measure supplies an invariant inner product on each finite-dimensional representation, including the Adjoint representation.

Real torus ( $T^{n} = R^{n} / Z^{n}$ )