I am considering the following set up:Let G be a finite group,let Rep(G) denote the category of finite dimensional representations over C. Let V,W be representations of G in Rep(G). One can define a bilinear form on Rep(G) or inner product in K0(Rep(G)) (in Teleman’s notes) as dimCHom(V,W)G which is G invariant of Hom(V,W).Then there … Read more
Hello to MathoverFlow community I have some seemingly technical questions on applications of geometric quantisation to Lie group representation theory. We shall start by giving background definitions. We follow [1]. Let (M,ω) be a quantizable symplectic manifold. Then there exists a Hermitian line bundle B→M and a connection ∇ on B with curvature ℏ−1ω. Let … Read more
I have a few questions with respect to Bezrukavnikov’s proof of the dimension formula for affine springer fibers in Fixed point set on affine flag manifolds. The setting is as follows: Let G be a reductive group over C. Let F=C((t)) with the valuation that satisfies νF(t)=1, and O=C[[t]]. Let N∈g(O) be a regular semisimple … Read more
Recall that a prehomogeneous vector space, is a representation $V$ of a linear algebraic group $G$ having an open $G$-orbit. Let $Z$ be the neutral connected component of the stabilizer of a point of the open orbit. Assume that $G$ and $Z$ are both reductive. Is it always true that the canonical homomorphism $Z/[Z,Z]\to G/[G,G]$ … Read more
I am interested in understanding Lemma A.2 in the paper “Moduli spaces of principal F-bundles” by varshavsky which you can find here. It uses so called “Plücker” coordinates of the flag variety for which I would like a reference. Let me explain the statement : Let G be a split reductive group over a field … Read more
From the standard results on Springer fibers of type A, we know that given a Springer fiber, say Bλ, its irreducible components are all equidimensional and parametrized by standard Young tableaux of the Young diagram associated to partition λ. Now the question: Given two standard Young Tableaux Tλ1 and Tλ2, of Young diagram λ, is … Read more
Diagonal coinvariants have an interpretation from https://arxiv.org/abs/math/0201148 in terms of the Hilbert scheme. There are two recent papers https://arxiv.org/pdf/1801.09033.pdf and https://arxiv.org/pdf/1808.02278.pdf which both detail (different and seemingly unconnected) relations between diagonal coinvariants and affine Springer fibers (another paper that does this is https://arxiv.org/pdf/1203.5878.pdf). In the first paper diagonal coinvariants act on suitable (co)homology of affine … Read more
Say everything is over $\mathbb C$, and I have an action $act: N \times X \to X$ of an affine algebraic group $N$ on a smooth variety, say with finitely many orbits. I’m trying to compute the $\mathcal D$-module direct image $act_+$ for sheaves on $N \times X$ of the form $\mathcal O_N \boxtimes \mathcal … Read more
Let G be a finite dimensional semisimple algebraic group, and for s∈W write is:Fs=B+sB−/B−→F for the sth Bruhat cell in the flag variety F. Then (in Affine Kac-Moody Algebras and Semi-Infinite Flag Manifolds bottom of p.165), Feigin-Frenkel claim that the local cohomology H∗Fs(F,O) = Mss⋆0 = Mssρ−ρ is an s–twisted Verma module (ignoring shifts). However, I’m pretty sure that … Read more
It is well known that for a vector space V with dim(V)=n+1 the GL(V)−module V⊗d splits as a sum of irreducible representations (with suitable multiplicities) SλV, where λ=(λ1,…,λr) is a partition of d (with suitable properties). For example in the case d=2 we have that V⊗V=Sym2(V)⊕2⋀(V) associated to partitions (2,0) and (1,1). Let Sn,d denote … Read more
