154 problems

Oliver's p-group conjecture.

Let $\mu$ be a splitting character used to define the oscillator–Heisenberg representation, and let…

Let $\chi=\chi^{\phi,\lambda}$ be a primary irreducible character of $\mathrm{GU}_n(q)$, where $\phi$ is an orbit other than $\{\mathbf{1}\}$ and $\mathbf{1}$ is the trivial charac…

Let $r+s=n$, with signature $(r,s)$, and let $M_I^{\mathrm{loc}}(r,s)$ be the scheme-theoretic closure of the generic fiber of the naive local model $M_I^{\mathrm{naive}}(r,s)$. Le…

Lapid–Mao's local conjecture. For every $\eta\in E_v^\times$ such that $\bar\eta=-\eta$, one has

Let $(\mathrm{G},b,\mu)$ be a PEL-type Rapoport–Zink datum, with $\mathrm{G}$ a connected reductive group over $\mathbb{Q}_p$, $\mu$ a minuscule cocharacter, and…

Let $S=S(G,h)$ be a Shimura variety with $G\otimes_{\mathbb{Q}}\mathbb{R}=GU(r,s)$, and let $M_{r,s}$ be the associated local model over $\operatorname{Spec}(R)$, where $R$ is the…

Smooth transfer conjecture. For any relative endoscopic datum $(\xi,\alpha,\beta)$ and any $f\in C_c^\infty(\mathfrak g_1)$, there exists…

Let $U$ be Haar-distributed on the unitary group and let $j\in\mathbb{Z}_{+}$. The normalized trace $\operatorname{Tr} U^j/\sqrt{j}$ converges in distribution to a standard normal…

Let $F$ be a $p$-adic field, let $E$ and $K$ be quadratic field extensions of $F$, and let $V$ be an $n$-dimensional skew-hermitian space relative to $E/F$. Set $V_K=V\otimes_F K$,…

Arithmetic transfer conjecture for odd type $(r,0)$. There exists $\varphi'\in C_c^\infty(G')$ transferring to $(c_r^2\\,\mathbf{1}_{K_n^{[r]}\times K_{n+1}},0)$ such that, for eve…

Let $\varphi'\in\mathcal{H}_{K'^{{{{{{{{\flat}}}}}}}}}\otimes\mathcal{H}_{K'}$, and let…

Let $p$ be an odd prime, let $\mathscr{W}$ be the rigid analytic space parametrizing continuous characters of $\mathbb{Z}_p^\times$, and write $T(w)=w(\exp(p))-1$ for the $T$-coord…

Let $G'$ be the relevant quasi-split unitary group, let $\phi'$ be an $L$-parameter of $G'$, and let $A_M$ be the finite group attached to the parameter as in the surrounding const…

Let $F$ be a CM field, let $\Sigma$ be a CM type, and let $F^+$ be the maximal totally real subfield of $F$. For each signature $I:\Sigma\to\{0,1,\ldots,n\}$, let $U_I$ be the corr…

Let $G_n$ and $G_{n+1}$ be the relevant unitary groups, with automorphic representations $\pi_n$ and $\pi_{n+1}$, global period $\mathcal{P}$, and normalized local periods…

Vertex-wise admissibility conjecture. Admissibility has a vertex-by-vertex characterization; equivalently,

Fulman's conjecture. As $n\to\infty$,

Let $E$ be the quadratic imaginary field in the paper, let $G_E$ be its absolute Galois group, and let $\pi$ be an automorphic representation of $\operatorname{U}(m)$ with regular…

Let $U_n$ be the finite unitary group, let $R$ be its space of class functions, and let $\chi^{\boldsymbol{\lambda}}\in R$ satisfy $\chi^{\boldsymbol{\lambda}}(1)>0$ and…

Let $E/F$ be a degree-two extension, let $G={\rm GL}_n(E)={\rm R}_{E/F}{\rm U}(n)$ in the notation of the source, and let $H={\rm U}(n)$. The pair $(G,H)$ is a supercuspidal Gelfan…

Let $F$ be a $p$-adic field, let $E/F$ be a degree-two extension, let ${\rm U}(n)$ be the corresponding unitary group, and let $\pi$ be a representation of ${\rm GL}_n(E)$. A repre…

Let $H$ be a Hilbert space, let $U(H)$ be its unitary group, and consider invariant means on $U(H)$. A multiplicative invariant mean is an invariant mean belonging to the multiplic…

Genericity transfer conjecture. If $\pi_1\otimes\pi_2$ is an irreducible subrepresentation of this function space and $\pi_1$ is generic, then $\pi_2$ is also generic.

Let $\Gamma=([n],w)$ be an arbitrary hypergraph with non-negative weights. Let $\mathcal{KMP}_k$ be the discrete KMP process with $k$ indistinguishable particles on $\Gamma$, and l…