73 problems

The affine Hecke algebra and the homomorphism referred to below are as in Cautis--Kamnitzer's construction; the statement concerns surjectivity before extending scalars to…

Let $\pi$ be a generic standard representation of a reductive $p$-adic group, and let $L$ denote its Langlands quotient. Standard module conjecture. The Langlands quotient $L$ is g…

Let $W$ be a Coxeter group with $r$ conjugacy classes of simple reflections, and identify weight functions on $W$ with points of $V=\mathbb{R}^{r}$ having integer coordinates, call…

Let $G$ be a reductive group with nilpotent cone $\mathfrak N$, let ${\mathfrak O}$ be a $G$-orbit in $\mathfrak N$, and let $\Omega$ be the corresponding two-sided Kazhdan–Lusztig…

Let $U_q(\widehat{sl}_{n+1})$ be the quantum affine algebra, and let $\widehat{H}_{\ell}(q^2)$ be the affine Hecke algebra. The Frobenius–Schur correspondence for…

Assume the crystal basis $B_\theta$ exists, and let $\widetilde F_i$ be its modified crystal operators. Distinctness conjecture. For every $b\in B_\theta$ and distinct $i,j\in I$,…

Assume that no $i\in I$ satisfies $\theta(i)=i$, and use the lattice $L_\theta$, its bar-conjugate $L_\theta^-$, and the integral module $(V_\theta)_\mathbf A$ defined from the bar…

Let $I=\{p_0p_1^{2n};n\in\mathbb Z\}\cup\{p_0^{-1}p_1^{2n};n\in\mathbb Z\}$, assume that no integer $n$ satisfies $p_0^2=p_1^{4n}$, and let $\theta(a)=a^{-1}$. Let $\mathfrak g_I$…

Let $I=\{p_1^n;n\in\mathbb Z_{\mathrm{odd}}\}$, assume that none of $\pm1$ and $\pm p_0$ lies in $I$, and let $\theta(a)=a^{-1}$. Let $\mathfrak g_I$ be the associated Lie algebra…

Let $\mathcal{H}$ be the affine Hecke algebra with labels $q_2=q_1^m$ obtained from the parameters $q_{\check{\alpha}}$ as in the source, with $q_1>1$ and $q_2\geq 1$. Let $W_0r$ b…

Let $G$ be a connected reductive algebraic group over ${\mathbb{C}}$ with simply-connected derived group, let $\tilde{W}_a$ be its extended affine Weyl group, and let…

Let $G$) be a connected reductive algebraic group over ${\mathbb{C}}$ with simply-connected derived group, let $\tilde{W}_a=WP$ be its extended affine Weyl group, and let…

Let $d$ and $p$ be positive integers, and let ${\bold M}_{\bold c}$ be the ${\mathbb Z}[q,q^{-1}]$-span of the alcoves in ${\Cal A}_{\bold c}$, viewed as a module for the affine He…

Haines–Opdam formula conjecture. The same formula holds for the full affine Hecke algebra.

Residual-degree rationality conjecture. The positive reals $d_\delta$ are rational numbers.

Fix $t=e^{2i\pi/p}$, where $p$ is the Coxeter number of the Coxeter–Dynkin diagram, and let $\rho_{hh'}$ be an irreducible representation occurring in the decomposition of the Hilb…

Let $\mathcal{H}$ be the Hecke algebra with standard basis $\{H_x\mid x\in W_{\mathrm{aff}}\}$, and let $\mathcal{B}\subseteq\mathcal{H}$ be the subalgebra under consideration. Let…

Positivity conjectures. The Laurent polynomials $Q_{x,w}\in\mathbb{Z}[v,v^{-1}]$ have non-negative coefficients for all $x\in W_{\mathrm{aff}}$ and $w\in W_{\mathrm{aff}}^+$, and t…

Assume $q\neq0$. Let $\operatorname{\mathcal{H}}_n^{\operatorname{aff}}$ be the affine Hecke algebra containing $\operatorname{\mathcal{H}}_n$ and the Laurent polynomial ring in…

The center and finite generation conjecture. We have

Let $\mathcal H_{\mathfrak s}$ be the affine Hecke algebra associated with a Bernstein block $\operatorname{Rep}(\mathcal G(F))^{\mathfrak s}$, where…

Let $\mathcal H$ be an affine Hecke algebra and let a generic standard representation mean a standard representation whose inducing data are generic, with generic irreducible const…

Let $\widetilde{G}$ be the metaplectic cover and $(\widetilde{J},\widetilde{\lambda})$ the type constructed in the paper. Let $\widetilde{M}$ be the associated Levi subgroup,…

There is a metaplectic spherical Whittaker function, denoted $\widetilde{\mathcal{W}}^{m} = \widetilde{\mathcal{W}}^{m}(\mathbf{y}; g;v)$, and components…

Let $Z_e^{red}$ be the quotient of $Z_e$ by its unipotent radical. Let ${\bf Z}_e\subset (Z_e^{red})^2$ be the algebraic variety of pairs of commuting elements in $Z_e^{red}$. Let…