Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 167 (Introduction to Geometric Representation Theory)

An affine algebraic group over $k$ is an affine variety $G$ equipped with multiplication $G\times G\to G$, inversion $G\to G$, and an identity element satisfying the group axioms, with multiplication and inversion both regular maps. Dually, the coordinate ring $k[G]$ is a commutative Hopf algebra: multiplication on $G$ induces the comultiplication $\Delta :k[G]\to k[G]\otimes k[G]$, inversion induces the antipode, and evaluation at the identity is the counit.

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Choose algebra generators $f_1,\dots ,f_r$ of the finitely generated coordinate ring $k[G]$. The right regular action is locally finite: writingshows that every right translate of $f_i$ lies in the finite span of the $f_{ij}$. Hence all the $f_i$ lie in some finite-dimensional translation-stable subspace $V\subseteq k[G]$.

This gives a rational representation $\rho :G\to \mathrm{GL}(V)$. If $g\in \ker \rho$, then $f_i(xg)=f_i(x)$ for every $i$ and $x$. Setting $x=e$ gives $f_i(g)=f_i(e)$ for every algebra generator, and therefore for every regular function on $G$. Regular functions separate closed points of an affine variety, so $g=e$. Thus $\rho$ is a faithful representation of an affine algebraic group.

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For $G={\mathbb{G}}_a$, write $k[G]=k[t]$. Under the left-right regular representation of an affine algebraic group,so the action factors through the difference map ${\mathbb{G}}_a\times {\mathbb{G}}_a\to {\mathbb{G}}_a$. The degree filtrationis stable and has trivial one-dimensional successive quotients. The module is indecomposable; every nonzero submodule contains a nonzero translation difference of lower degree and, on iteration using suitable translations, meets the unique invariant line $k\cdot 1$.

For every finite-dimensional rational $G$-module $V$, the matrix-coefficient construction givesOn the other hand,Indeed, a nonzero finite-dimensional quotient of $k[G]$ would have a simple quotient. Every simple rational representation of the unipotent algebraic group ${\mathbb{G}}_a$ is trivial, so this would give a nonzero translation-invariant functional on $k[t]$. No such functional exists: if $c_j$ is its first nonzero value on a monomial, translating a sufficiently high monomial produces a nonzero coefficient times $c_j$, contradicting invariance.

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For $G={\mathbb{G}}_m$, write $k[G]=k[t,t^{-1}]$. Sincethe weight-space decomposition isas a $G\times G$-representation.

If $V={⨁}_r{V}_r$ is the weight decomposition of a finite-dimensional $G$-module, the matrix-coefficient argument again gives ${\mathrm{Hom}}_G(V,k[G])\cong V^{\ast }$. A map from $k[G]$ to $V$ is independently determined by the image in $V_r$ of the basis vector $t^r$; only finitely many $V_r$ are nonzero. Thusas vector spaces.

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An element $g\in G(k)$ is semisimple when its image in a faithful finite-dimensional representation is diagonalizable, and unipotent when that image has every eigenvalue equal to one. These conditions are independent of the faithful representation. The Jordan decomposition in an affine algebraic group is the unique factorizationwith $g_s$ semisimple and $g_u$ unipotent.

In $G={\mathrm{GL}}_2$, matrices with two distinct eigenvalues form a dense open subset and are semisimple, so semisimple elements are dense. Their full locus is not open: a scalar matrix is semisimple, but every neighbourhood of it contains a nontrivial Jordan block. The group also has nonidentity unipotent matrices.

For a contrasting example take $G={\mathbb{G}}_a\times {\mathbb{G}}_m$. Its semisimple locus is $\{0\}\times {\mathbb{G}}_m$, which is closed and not dense, while $(a,1)$ for $a\ne 0$ is unipotent and $(0,t)$ for $t\ne 1$ is semisimple.

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The derived subgroup of an affine algebraic group $[G,G]$ is the closed subgroup generated by the commutators $xy{x}^{-1}{y}^{-1}$. If $G$ is connected, the image of every finite product of commutator maps is connected and contains the identity. The closures of these images form an increasing sequence; once their dimensions stabilize, the stable member is closed under products and inverses and equals $[G,G]$. Hence $[G,G]$ is connected.

Now suppose the connected group $G$ is solvable. The Lie-Kolchin theorem conjugates a faithful representation of $G$ into the upper triangular matrices. Every commutator then has all diagonal entries equal to one, so every element of $[G,G]$ is unipotent. Moreover $[G,G]$ lies in the upper unitriangular group, whose superdiagonal filtration is a central series. It is therefore a nilpotent group.

A diagonalizable algebraic group is a closed subgroup of a product of copies of ${\mathbb{G}}_m$. A unipotent algebraic group has only unipotent elements, while a semisimple algebraic group here means one all of whose elements are semisimple. A reductive algebraic group is smooth, connected, affine, and has trivial connected normal unipotent radical.

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The Kolchin theorem conjugates a faithful representation of a unipotent algebraic group $U$ into the upper unitriangular group $U_n$. Let $U_n^{(r)}$ consist of matrices whose first $r-1$ superdiagonals vanish. Matrix multiplication givesSince $U_n^{(n)}=1$, this filtration is a finite central series, so $U_n$ and every subgroup of it are nilpotent groups. Hence $U$ is nilpotent.

The converse fails: ${\mathbb{G}}_m$ is abelian, hence nilpotent, but its nonidentity points are semisimple rather than unipotent.

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Let $I=I(H)\subseteq k[G]$. Choose finitely many generators of $I$ and a finite-dimensional $G$-submodule $U\subseteq k[G]$ containing them, using local finiteness of the right regular action. Set $W=U\cap I$. Right translation by $H$ preserves $I$, so $H$ stabilizes $W$. Conversely, if $gW=W$, every chosen generator $f$ has $R_{g}f\in I$, and evaluation at the identity gives $f(g)=0$. Thus $g\in H$, and ${\mathrm{Stab}}_G(W)=H$.

Put $d=\dim W$ and take $V={\bigwedge }^{d}U$. The linedetermines $W$ uniquely, so its stabilizer is the stabilizer of $W$, namely $H$.

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A flat algebraic-group torsor $X\to Y$ is a faithfully flat morphism with a right $H$-action for whichis an isomorphism.

The orbit $G\ell$ is a locally closed subvariety of $\mathbb{P}(V)$ by the orbit theorem for algebraic-group actions. The fibers of the orbit map $G\to G\ell$ are precisely the right cosets of $H={\mathrm{Stab}}_G(\ell )$, and the displayed action map is therefore an isomorphism. The theorem on quotients of affine algebraic groups by closed subgroups says that $G/H$ exists and $G\to G/H$ is faithfully flat; the induced map $G/H\to G\ell$ is an isomorphism. Hence the orbit map is a flat $H$-torsor.

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For $G={\mathrm{SL}}_2$ and the upper triangular Borel subgroup $B$, the quotient is $G/B\cong {\mathbb{P}}^1$: a matrix is sent to the line spanned by its first column. Over $U_0=\{[1:t]\}$ use the sectionand over $U_1=\{[s:1]\}$ useEvery matrix above $U_i$ is uniquely $s_i(u)b$ with $b\in B$. Thus each inverse image is $U_i\times B$, proving that $G\to G/B$ is a Zariski $B$-torsor.

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For $G={\mathbb{G}}_m$ and $H={\mu }_2$, the quotient isIt is a faithfully flat ${\mu }_2$-torsor, including in characteristic two where ${\mu }_2$ is nonreduced. It is not a Zariski torsor: a local section over any nonempty open set would put a square root of the coordinate $t$ in the function field $k(t)$, but $t$ is not a square there.

Take $V=k\oplus k$ with $z\in {\mathbb{G}}_m$ acting by $z\cdot (x,y)=(x,z^{2}y)$, and let $\ell =k(1,1)$. The equality $z\ell =\ell$ holds exactly when $z^2=1$, so the scheme-theoretic stabilizer is ${\mu }_2$.

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Embed the space $\mathcal{B}$ of complete isotropic flags intoThe incidence conditions $F_i\subset F_{i+1}$ and isotropy equations $⟨F_i,F_i⟩=0$ are closed polynomial conditions. Since the Grassmannian is projective, $\mathcal{B}$ is a projective algebraic variety.

Every complete isotropic flag extends to a symplectic basis. A symplectic change of basis carries any such flag to any other, so $G={\mathrm{Sp}}_{2n}$ acts transitively. The stabilizer $B$ of the standard flag consists of the upper triangular symplectic matrices. It is closed, connected, and solvable. The Lie-Kolchin theorem shows that every connected solvable subgroup fixes a complete flag in $V$; preservation of the symplectic form makes the resulting flag isotropic after taking its first half. Such a subgroup is conjugate into $B$, so $B$ is maximal and hence a Borel subgroup.

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Choose the maximal torusand write ${\epsilon }_i(t)=t_i$. Thenwhere the type-$C_n$ root system isHere $X^{\ast }(T)={⨁}_i\mathbb{Z}{\epsilon }_i$ and $X_{\ast }(T)={⨁}_i\mathbb{Z}{\epsilon }_i^{\vee }$. The root datum haswith the corresponding negatives.

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The normalizer of $T$ permutes the $n$ symplectic coordinate planes and may interchange the two basis vectors in each plane. Modulo $T$, these operations give every signed permutation of $({\epsilon }_1,\dots ,{\epsilon }_n)$. Thereforethe signed symmetric group, generated by adjacent coordinate transpositions and one sign change.

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For each signed permutation $w\in W$, choose its symplectic signed-permutation matrix $\dot{w}\in N_G(T)$. The Bruhat decomposition of a reductive algebraic group is the explicit disjoint union

To prove existence, compare the standard isotropic flag $F_{\bullet }$ with $g{F}_{\bullet }$. The rankstogether with the symplectic orthogonality relations determine a unique signed permutation $w$. Symplectic row and column operations from $B$ then reduce $g$ to $\dot{w}$, so $g\in B\dot{w}B$. Conversely, the same intersection dimensions are constant on a double coset and recover $w$, proving disjointness. This is symplectic Gaussian elimination and establishes the claimed decomposition.

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Put $i_0=0$ and $d_j=i_j-i_{j-1}$. The stabilizeris the standard parabolic subgroup containing $B$ obtained by allowing arbitrary changes of basis inside the successive blocks. Its Levi subgroup isThe fiber of ${\pi }_I$ over the displayed partial flag is $P_I/B$. Choosing a complete refinement amounts to choosing complete flags in every quotient $F_{i_j}/F_{i_{j-1}}$ and a complete isotropic flag in $F_{i_r}^{\perp }/F_{i_r}$. Hencethe flag variety of an algebraic group $L_I$; explicitly it is the product of the complete flag varieties of the listed general linear and symplectic factors.

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The center consists of scalar symplectic transformations:This description retains the nonreduced center in characteristic two. Since $Z_G$ is a finite central subgroup scheme, the invariant ring $k[G{]}^{Z_G}$ is finitely generated andis an affine algebraic group; the quotient map is finite and faithfully flat.

The quotient torus is $T/{\mu }_2$. Its character and cocharacter lattices areThe roots and coroots are the same type-$C_n$ sets written in part (ii), now regarded in these lattices. This is the adjoint root datum of type $C_n$.

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