Let G→X be a parahoric group scheme over a curve, with parahoric level structure at x0. Gaitsgory essentially showed that the nearby cycles functor RΨ takes perverse sheaves on the affine Grassmannian GrG|X−x0 to central perverse sheaves on the affine flag variety GrG|x0=FlG. This shows that the Frobenius trace function is central in the parahoric … Read more
Does there exist a homogenous conjugate-linear automorphism of the KLR algebra? I want to be able to define a homogenous Hermitian form on the (Specht) modules of the (cyclotomic) KLR algebra. Answer AttributionSource : Link , Question Author : Chris Bowman , Answer Author : Community
My apologies if the below is too malformed to make sense. Let (W,S) be the affine Weyl group of a reductive group G, and let {Cw} be the Kazhdan-Lusztig C-basis (an answer in terms of the C′w-basis is of course equally useful, though). Let H be the corresponding Hecke algebra over Z[q,q−1], and write hx,y,z … Read more
This is a spin-off to the question Omitting primes from a Hecke algebra by David Loeffler. Let N be a positive integer. For a finite set of primes Σ, let TΣ be the Z-subalgebra of endomorphisms of S2(Γ1(N)) generated by Hecke operators Tℓ for all prime ℓ∉Σ. If ℓ∈Σ implies that ℓ∤, then it is … Read more
I am reading Chriss, Ginzburg’s book Representation theory and complex geometry. In theorem 7.2.16 it says that the convolution action of the Steinberg variety $St=\tilde{\mathcal{N}}\times_\mathfrak{g}\tilde{\mathcal{N}}$ on the Springer resolution $\tilde{\mathcal{N}}$ induces an action on $K$-groups, i.e. an action of $K_0^{G\times\mathbb{G}_m}(St)=H_\text{aff}$ on $K_0^{G\times\mathbb{G}_m}(\tilde{\mathcal{N}})=R(T)$. Then the claim is that the element $T_{s_\alpha}\in H_\text{aff}$ acts as $$T_{s_\alpha}:e^\lambda\mapsto\frac{e^\lambda-e^{s_\alpha(\lambda)}}{e^{\alpha}-1}-q\frac{e^\lambda-e^{s_\alpha(\lambda)+\alpha}}{e^{\alpha}-1}$$ where … Read more
It is shown by Berest-Etingof-Ginzburg that there exist finite-dimensional irreducible representations of rational Cherednik algebra $H_c(S_n)$ of $A_{n-1}$ type if and only if the deformation parameter $c$ takes the rational numbers of the form $c=m/n$. Since the rational Cherednik algebra is the rational degeneration of double affine Hecke algebra (DAHA), the question is as follows: … Read more
From the Eichler-Shimura relation, we have a formula for $T_p$ when we reduce $\textrm{End}(\textrm{Jac}(X))$ mod $p$. Explicity, $T_p=\textrm{Frob}_p+p\textrm{Frob}_p^{-1}$. Is there a way to define the Hecke operator as a lift of this operator satisfying certain other properties? Is there a definition of $T_p$ which does not rely on a moduli space interpretation or double coset … Read more
I was reading James Cogdell’s notes here on automorphic representations and came to the following claim about the spherical Hecke algebra H(GL2(Qp),GL2(Zp)). He remarks that this Hecke algebra identifies with the algebra spanned by the classical Hecke operators Tpr. I have read about Hecke operators in Diamond and Shurman’s book on modular forms, but have … Read more
I am searching for a book/lecture notes/articles where I can find the definition and properties of the pro−p Iwahori subgroup of GLn(F),(with examples if possible) the Iwahori decomposition of it, and its applications concerning the Hecke Algebra. The subgroups play an important role in the work of M.F. Vigneras as evident from the appendix of … Read more
I’m looking for examples of non-trivial Kazhdan-Lusztig polynomials, specifically in the case where the Coxeter system is a Weyl group. For example, the simplest polynomial with non-trivial q-coefficient is ptsut,e(q)=1+q in type A3. Where can we find the first non-trivial coefficient of q2, and q3… etc. Answer Already in the case of finite symmetric groups, … Read more
