42 problems

Let $H=H^{\mathrm{SL}_n}$ be the maximal torus of $SL_n(\mathbb C)$, let $\vartheta\in\mathfrak h^*/W$, and set $k=c-1$. Let $\nabla^{K}(\vartheta,k)$ be the trigonometric KZ conne…

Let $\Gamma_{J,\mathsf{v}}$ be the cell recognised by the bounded representation $(\pi_{J,\mathsf{v}},M_{J,\mathsf{v}},\mathsf{B}_{J,\mathsf{v}})$, let $\mathbf{a}_{J,\mathsf{v}}$…

Product formula conjecture. If the representation is bounded, then

Let $(\pi_{J,\mathsf{v}},M_{J,\mathsf{v}},\mathsf{B}_{J,\mathsf{v}})$ be a bounded representation, let $\mathbf{a}_{J,\mathsf{v}}$ denote its bound, let $J$ be the relevant subset…

Let $I=\{p^n\mathbin{;}n\in\mathbb Z_{\mathrm{odd}}\}$, with neither $1$ nor $-1$ in $I$, and let $\theta(a)=a^{-1}$. Let $K_{D,I}$ be the Grothendieck group of finite-dimensional…

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 $I$ be the parameter set with involution $\theta$, and assume that no $i\in I$ satisfies $\theta(i)=i$. Let $V_\theta$ be the type-D module constructed in the source, with oper…

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…

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…

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,…

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…

Let ${\bf H}=S_c^\infty(G)$ be the algebra of locally constant compactly supported functions on the $p$-adic group, let ${\bf J}$ be the corresponding algebra, and let…

Let $\tilde{W}$ be an affine or extended affine Weyl group for which Lusztig's denominator conjecture holds, and let $P_W$ be the Poincaré polynomial of its finite Weyl group. Fix…

Lusztig's denominator conjecture. For all $x,w\in W_{\mathrm{aff}}$, $a_{x,w}$ is a rational function of $\mathbf{q}$; its denominator is independent of $x$ and, as a function of…

Let $\Phi$ be a root system of type $A_n$, let $X^+$ denote the dominant weights, and let $\mathbf{N}^i$ be the pre-canonical bases of the spherical Hecke algebra. Positivity conje…

Let $C\rightsquigarrow C_0$ be a degeneration of a smooth complex projective curve to a nodal curve, with marked points $S\rightsquigarrow S_0$ disjoint from the nodes. Let…

Let $ccalC_0$ and $ccalB_0$ be the source and target categories, let $ccalM:ccalC_0\longrightarrow ccalB_0$ be the sheaf-theoretic functor, and let $cPhi_*$, $cKZ$, and $cPsi^*$ be…

Let $cKZ$ be the functor from the relevant category of modules over the affine Hecke algebra to the corresponding category of modules over the algebra $cbbK$. A projective object m…

Let the inner product be the construction given in Theorem, and let an affine type mean the type of an affine Weyl group or affine Hecke algebra. Inner product generalisation conje…

Let $\Pi$ be a class of representations appearing in the Plancherel theorem, and let $\Omega$ be the map from such classes to two-sided cells; write $\Gamma=\Omega(\Pi)$. Let…

Let $\Pi$ range over the classes of representations appearing in the Plancherel formula, and let the two-sided cells be the cells of the affine Hecke algebra. A Plancherel-to-cell…

An affine Hecke algebra is an algebra associated with an affine Weyl group and a choice of parameters; a balanced system of cell representations is a system of representations sati…