Generic cuspidal representations of 𝑈(2, 1)

Let 𝐹 be a non-Archimedean local field, and let 𝜎 be a non-trivial Galois involution with fixed field F 0 F_{0} . When the residue characteristic of F 0 F_{0} is odd, using the construction of cuspidal representations of classical groups by Stevens, we classify generic cuspidal representations of U ⁢ ( 2 , 1 ) ⁢ ( F / F 0 ) U(2,1)(F/F_{0}) .

Cuspidal ℓ-modular representations of p-adic classical groups

For a classical group over a non-archimedean local field of odd residual characteristic p, we construct all cuspidal representations over an arbitrary algebraically closed field of characteristic different from p, as representations induced from a cuspidal type. We also give a fundamental step towards the classification of cuspidal representations, identifying when certain cuspidal types induce to equivalent representations; this result is new even in the case of complex representations. Finally, we prove that the representations induced from more general types are quasi-projective, a crucial tool for extending the results here to arbitrary irreducible representations.

Burgess-like subconvexity for

We generalize our previous method on the subconvexity problem for $\text{GL}_{2}\times \text{GL}_{1}$ with cuspidal representations to Eisenstein series, and deduce a Burgess-like subconvex bound for Hecke characters, that is, the bound $|L(1/2,\unicode[STIX]{x1D712})|\ll _{\mathbf{F},\unicode[STIX]{x1D716}}\mathbf{C}(\unicode[STIX]{x1D712})^{1/4-(1-2\unicode[STIX]{x1D703})/16+\unicode[STIX]{x1D716}}$ for varying Hecke characters $\unicode[STIX]{x1D712}$ over a number field $\mathbf{F}$ with analytic conductor $\mathbf{C}(\unicode[STIX]{x1D712})$ . As a main tool, we apply the extended theory of regularized integrals due to Zagier developed in a previous paper to obtain the relevant triple product formulas of Eisenstein series.

On Twisted Gelfand Pairs Through Commutativity of a Hecke Algebra

For a locally compact, totally disconnected group $G$, a subgroup $H$, and a character $\chi :H \to \mathbb{C}^{\times }$ we define a Hecke algebra ${\mathcal{H}}_\chi$ and explore the connection between commutativity of ${\mathcal{H}}_\chi$ and the $\chi$-Gelfand property of $(G,H)$, that is, the property $\dim _{\mathbb{C}} (\rho ^*)^{(H,\chi ^{-1})} \leq 1$ for every $\rho \in \textrm{Irr}(G)$, the irreducible representations of $G$. We show that the conditions of the Gelfand–Kazhdan criterion imply commutativity of ${\mathcal{H}}_\chi$ and verify in several simple cases that commutativity of ${\mathcal{H}}_\chi$ is equivalent to the $\chi$-Gelfand property of $(G,H)$. We then show that if $G$ is a connected reductive group over a $p$-adic field $F$, and $G/H$ is $F$-spherical, then the cuspidal part of ${\mathcal{H}}_\chi$ is commutative if and only if $(G,H)$ satisfies the $\chi$-Gelfand property with respect to all cuspidal representations ${\rho \in \textrm{Irr}(G)}$. We conclude by showing that if $(G,H)$ satisfies the $\chi$-Gelfand property with respect to all irreducible $(H\backslash G,\chi ^{-1})$-tempered representations of $G$ then ${\mathcal{H}}_\chi$ is commutative.