Intertwining maps between 𝑝-adic principal series of 𝑝-adic groups

In this paper we study p p -adic principal series representation of a p p -adic group G G as a module over the maximal compact subgroup G 0 G_0 . We show that there are no non-trivial G 0 G_0 -intertwining maps between principal series representations attached to characters whose restrictions to the torus of G 0 G_0 are distinct, and there are no non-scalar endomorphisms of a fixed principal series representation. This is surprising when compared with another result which we prove: that a principal series representation may contain infinitely many closed G 0 G_0 -invariant subspaces. As for the proof, we work mainly in the setting of Iwasawa modules, and deduce results about G 0 G_0 -representations by duality.

R-GROUP AND WHITTAKER SPACE OF SOME GENUINE REPRESENTATIONS

For a unitary unramified genuine principal series representation of a covering group, we study the associated R-group. We prove a formula relating the R-group to the dimension of the Whittaker space for the irreducible constituents of such a principal series representation. Moreover, for certain saturated covers of a semisimple simply connected group, we also propose a simpler conjectural formula for such dimensions. This latter conjectural formula is verified in several cases, including covers of the symplectic groups.

An invitation to the principal series

Scalar unitary representations of the isometry group of dd-dimensional de Sitter space SO(1,d)SO(1,d) are labeled by their conformal weights \DeltaΔ. A salient feature of de Sitter space is that scalar fields with sufficiently large mass compared to the de Sitter scale 1/\ell1/ℓ have complex conformal weights, and physical modes of these fields fall into the unitary continuous principal series representation of SO(1,d)SO(1,d). Our goal is to study these representations in d=2d=2, where the relevant group is SL(2,\mathbb{R})SL(2,ℝ). We show that the generators of the isometry group of dS_22 acting on a massive scalar field reproduce the quantum mechanical model introduced by de Alfaro, Fubini and Furlan (DFF) in the early/late time limit. Motivated by the ambient dS_22 construction, we review in detail how the DFF model must be altered in order to accommodate the principal series representation. We point out a difficulty in writing down a classical Lagrangian for this model, whereas the canonical Hamiltonian formulation avoids any problem. We speculate on the meaning of the various de Sitter invariant vacua from the point of view of this toy model and discuss some potential generalizations.

Yang-Baxter basis of Hecke algebra and Casselman's problem (extended abstract)

International audience We generalize the definition of Yang-Baxter basis of type A Hecke algebra introduced by A.Lascoux, B.Leclerc and J.Y.Thibon (Letters in Math. Phys., 40 (1997), 75–90) to all the Lie types and prove their duality. As an application we give a solution to Casselman's problem on Iwahori fixed vectors of principal series representation of p-adic groups.

Highest Weights for Categorical Representations

We present a criterion for establishing Morita equivalence of monoidal categories and apply it to the categorical representation theory of reductive groups $G$. We show that the “de Rham group algebra” $\mathcal D(G)$ (the monoidal category of $\mathcal D$-modules on $G$) is Morita equivalent to the universal Hecke category $\mathcal D({N}\backslash{G}/{N})$ and to its monodromic variant $\widetilde{\mathcal D}({B}\backslash{G}/{B})$. In other words, de Rham $G$-categories, that is, module categories for $\mathcal D(G)$, satisfy a “highest weight theorem”—they all appear in the decomposition of the universal principal series representation $\mathcal D(G/N)$ or in twisted $\mathcal D$-modules on the flag variety $\widetilde{\mathcal D}(G/B)$.

Fourier supports of K-finite Bessel integrals on classical tube domains

Let [Formula: see text] be the symmetric tube domain associated with the Jordan algebra [Formula: see text], [Formula: see text], [Formula: see text], or [Formula: see text], and [Formula: see text] be its Shilov boundary. Also, let [Formula: see text] be a degenerate principal series representation of [Formula: see text]. Then we investigate the Bessel integrals assigned to functions in general [Formula: see text]-types of [Formula: see text]. We give individual upper bounds of their supports, when [Formula: see text] is reducible. We also use the upper bounds to give a partition for the set of all [Formula: see text]-types in [Formula: see text], that turns out to explain the [Formula: see text]-module structure of [Formula: see text]. Thus, our results concretely realize a relationship observed by Kashiwara and Vergne [[Formula: see text]-types and singular spectrum, in Noncommutative Harmonic analysis, Lecture Notes in Mathematics, Vol. 728 (Springer, 1979), pp. 177–200] between the Fourier supports and the asymptotic [Formula: see text]-supports assigned to [Formula: see text]-submodules in [Formula: see text].

A QUOTIENT OF THE LUBIN–TATE TOWER

In this article we show that the quotient${\mathcal{M}}_{\infty }/B(\mathbb{Q}_{p})$of the Lubin–Tate space at infinite level${\mathcal{M}}_{\infty }$by the Borel subgroup of upper triangular matrices$B(\mathbb{Q}_{p})\subset \operatorname{GL}_{2}(\mathbb{Q}_{p})$exists as a perfectoid space. As an application we show that Scholze’s functor$H_{\acute{\text{e}}\text{t}}^{i}(\mathbb{P}_{\mathbb{C}_{p}}^{1},{\mathcal{F}}_{\unicode[STIX]{x1D70B}})$is concentrated in degree one whenever$\unicode[STIX]{x1D70B}$is an irreducible principal series representation or a twist of the Steinberg representation of$\operatorname{GL}_{2}(\mathbb{Q}_{p})$.

THE DOUBLE COVER OF ODD GENERAL SPIN GROUPS, SMALL REPRESENTATIONS, AND APPLICATIONS

We construct local and global metaplectic double covers of odd general spin groups, using the cover of Matsumoto of spin groups. Following Kazhdan and Patterson, a local exceptional representation is the unique irreducible quotient of a principal series representation, induced from a certain exceptional character. The global exceptional representation is obtained as the multi-residue of an Eisenstein series: it is an automorphic representation, and it decomposes as the restricted tensor product of local exceptional representations. As in the case of the small representation of$\mathit{SO}_{2n+1}$of Bump, Friedberg, and Ginzburg, exceptional representations enjoy the vanishing of a large class of twisted Jacquet modules (locally), or Fourier coefficients (globally). Consequently they are useful in many settings, including lifting problems and Rankin–Selberg integrals. We describe one application, to a calculation of a co-period integral.