Euler characteristic, character of group representation and Riemann Roch theorem

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

Reference request: Prequantization of canonical transformations and Lie group action

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

Dimension of affine springer fiber

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

Prehomogeneous vector spaces for reductive groups

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

Plucker coordinates of flag varieties

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

Intersection of components in Springer fibre of type A

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

Relations between double coinvariants and affine Springer fibers

Diagonal coinvariants have an interpretation from in terms of the Hilbert scheme. There are two recent papers and which both detail (different and seemingly unconnected) relations between diagonal coinvariants and affine Springer fibers (another paper that does this is In the first paper diagonal coinvariants act on suitable (co)homology of affine … Read more

Computing $\mathcal D$-module direct image along group action map

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

Confusion about twisted Vermas in Feigin-Frenkel

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

Tensors of minimal rank in Schur modules SλV⊂V⊗|λ|S_{\lambda}V \subset V^{\otimes |\lambda|}

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