Semisimple vs ordinary trace of Frobenius on nearby cycles of affine flag variety

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

Kazhdan-Lusztig basis elements appearing in product with distinguished involution

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

Index of the Hecke algebra with operators omitted

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

A problem on Kazhdan–Lusztig theorem

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

Finite-dimensional representations of DAHA

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

Is there a way to define Hecke operators “inherently” as certain endomorphisms of the Jacobian?

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

Hecke algebra H(GL2(Qp)/GL2(Zp))\mathcal H(\operatorname{GL}_2(\mathbb Q_p)/\operatorname{GL}_2(\mathbb Z_p)) and Hecke operators

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

Reference request for pro−ppro-p Iwahori subgroup of GLn(F)GL_n(F)

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

Examples of non-trivial Kazhdan-Lusztig polynomials

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