17 problems

Vogan's unipotent representation conjecture. Unipotent representations are unitary and form the building blocks of the unitary dual.

Kernel Unipotent Conjecture. The Zassenhaus filtration of $G_K$ is given by the kernels of unipotent representations.

Reeder's rational-ratio conjecture. The formal dimensions of the unipotent discrete series representations of $\mathcal{G}$ within one unipotent discrete series $L$-packet are prop…

Let $G$ be a reductive group over a non-archimedean local field $F$, let $\pi$ be a unipotent irreducible representation of $G(F)$, and let $\phi$ be its $L$-parameter. Say that…

Let $G=\mathrm G(F)$, where $\mathrm G$ is a connected reductive algebraic group defined over $F$ and inner to a split group. Let ${}^{K}\mathrm{WF}(\pi)$ denote the unramified can…

Let $U(p,q)$ be a real unitary group, and let a special unipotent representation mean a representation in the special-unipotent class considered in the paper. A representation is f…

Sommers' conjecture. The set $^K\WF(X)$ is a singleton and

Cuspidality conjecture. Then $p$ is $\tau_d$-cuspidal.

Unipotent-family conjecture. There exists a unique bijection

Consider the unipotent representation … of complex $E_7$, whose $K$-types have highest weights in the weight lattice but not in the root lattice and occur with multiplicity one. Mc…

Let $G$ be a simple $F$-split group. Let ${\mathcal R}^p_{{\mathrm{un}},{\mathrm{ell}}}(G)$ be the elliptic unipotent representation space, let…

Let $\mathfrak{g}$ be the Lie algebra of a complex semisimple Lie group. Let $\mathcal{O}$ be a special nilpotent orbit, let $\overline{A}(\mathcal{O})$ be its Lusztig quotient, an…

Let $F$ be a field containing a primitive $p$-th root of unity, and let $G=G_F(p)$ be the maximal pro-$p$ Galois group of $F$. For a pro-$p$ group $G$, the kernel $n$-unipotent pro…

Kernel $n$-unipotent conjecture. For every integer $n\geq 3$, the group $G_F(p)$ has the kernel $n$-unipotent property.

Let $q$ be a prime power, let $G=GL_n(q)$, let $U=U_n(q)$ be the lower unitriangular subgroup, and let $\lambda$ be a partition of $n$. Let $S_F^\lambda$ be the unipotent Specht mo…

Let $q$ be a prime power, let $G=GL_n(q)$, let $U\leqslant G$ be the subgroup of lower unitriangular matrices, and let $\lambda$ be a partition of $n$. Write $S^\lambda$ for the co…

Finiteness conjecture. The set of points of $G(K_0)_{rsc}$ at which a unipotent almost character takes the value $\infty$ is empty.