Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 19J / b

WriteSince $u(x)u(y)=u(x+y)$,Thus $\chi (v\cdot )$ is a one-dimensional representation of $U$; for $v=0$ it is the trivial representation.

For $a\in {\mathbb{F}}_p^{\ast }$ putConjugation givesSuppose the $\chi$-weight spaceis nonzero. If $0\ne z\in V_{\chi }$, then $t(a)z$ lies in the $\chi (a^{-2}\cdot )$-weight space, becauseHence every character indexed by a nonzero square occurs in ${\mathrm{Res}}_U^{B}V$.

These characters are distinct. Indeed, if $\chi (v\cdot )=\chi (v^{\prime }\cdot )$ with $v\ne v^{\prime }$, multiplication by $v-v^{\prime }$ would show that $\chi$ is trivial on all of $U$, contrary to hypothesis. There are $(p-1)/2$ nonzero squares, sofor at least $(p-1)/2$ values of $v\in {\mathbb{F}}_p^{\ast }$. This is the square orbits of additive characters of a finite field argument.

PutSince $B=U⋊T$ and $U\cap T=1$, restriction to $U$ gives its regular representation:As $U\cong ({\mathbb{F}}_p,+)$ is abelian, this is the direct sum of all $p$ one-dimensional characters of $U$, each with multiplicity one.

The diagonal subgroup $T$ permutes these one-dimensional weight spaces. It fixes the trivial character, while part (ii) shows that the nontrivial characters split into two orbits, indexed by the nonzero squares and nonsquares. Let $W_0,W_{+},W_{-}$ be the sums of the weight spaces in these three orbits. Thenas $B$-representations, and

Each summand is irreducible. Any nonzero $B$-subrepresentation decomposes into $U$-weight spaces. If it contains one of the one-dimensional weight spaces in an orbit, transitivity of the $T$-action forces it to contain every weight space in that orbit, hence the whole corresponding $W_0$, $W_{+}$, or $W_{-}$. The three summands are pairwise nonisomorphic because their restrictions to $U$ have disjoint character supports. Therefore the induction from the diagonal subgroup of upper-triangular SL2 decomposition has the required three irreducible constituents of dimensions