Compatible intertwiners for representations of finite nilpotent groups

We sharpen the orbit method for finite groups of small nilpotence class by associating representations to functionals on the corresponding Lie rings. This amounts to describing compatible intertwiners between representations parameterized by an additional choice of polarization. Our construction is motivated by the theory of the linearized Weil representation of the symplectic group. In particular, we provide generalizations of the Maslov index and the determinant functor to the context of finite abelian groups

Finite quotients of symplectic groups vs mapping class groups

We give alternative computations of the Schur multiplier of $Sp(2g,\mathbb Z/D\mathbb Z)$, when $D$ is divisible by 4 and $g\geq 4$: using K-theory arguments based on the work of Barge and Lannes and a second one based on the Weil representations of symplectic groups arising in abelian Chern-Simons theory. We can also retrieve this way Deligne's non-residual finiteness of the universal central extension $\widetilde{Sp(2g,\mathbb Z)}$. We prove then that the image of the second homology into finite quotients of symplectic groups over a Dedekind domain of arithmetic type are torsion groups of uniformly bounded size. In contrast, quantum representations produce for every prime $p$, finite quotients of the mapping class group of genus $g\geq 3$ whose second homology image has $p$-torsion. We further derive that all central extensions of the mapping class group are residually finite and deduce that mapping class groups have Serre's property $A_2$ for trivial modules, contrary to symplectic groups. Eventually we compute the module of coinvariants $H_2(\mathfrak{sp}_{2g}(2))_{Sp(2g,\mathbb Z/2^k\mathbb Z)}=\mathbb Z/2\mathbb Z$.Comment: 40p., 3 figures, former arxiv:1103.1855 is now split into two separate papers, the actual arxiv:1103.1855 and the present on

The Schur multiplier of finite symplectic groups

We show that the Schur multiplier of $Sp(2g,\mathbb Z/D\mathbb Z)$ is $\mathbb Z/2\mathbb Z$, when $D$ is divisible by 4.Comment: Bull. Soc. Math. France, to appea

Small representations of finite classical groups

Finite group theorists have established many formulas that express interesting properties of a finite group in terms of sums of characters of the group. An obstacle to applying these formulas is lack of control over the dimensions of representations of the group. In particular, the representations of small dimensions tend to contribute the largest terms to these sums, so a systematic knowledge of these small representations could lead to proofs of important conjectures which are currently out of reach. Despite the classification by Lusztig of the irreducible representations of finite groups of Lie type, it seems that this aspect remains obscure. In this note we develop a language which seems to be adequate for the description of the "small" representations of finite classical groups and puts in the forefront the notion of rank of a representation. We describe a method, the "eta correspondence", to construct small representations, and we conjecture that our construction is exhaustive. We also give a strong estimate on the dimension of small representations in terms of their rank. For the sake of clarity, in this note we describe in detail only the case of the finite symplectic groups.Comment: 18 pages, 9 figures, accepted for publications in the proceedings of the conference on the occasion of Roger Howe's 70th birthday (1-5 June 2015, Yale University, New Haven, CT