Coordinate rings and birational charts

Let G G be a semisimple simply connected complex algebraic group. Let U U be the unipotent radical of a Borel subgroup in G G . We describe the coordinate rings of U U (resp., G / U G/U , G G ) in terms of two (resp., four, eight) birational charts introduced by Lusztig [Total positivity in reductive groups, Birkhäuser Boston, Boston, MA, 1994; Bull. Inst. Math. Sin. (N.S.) 14 (2019), pp. 403–459] in connection with the study of total positivity.

Typical representations via fixed point sets in Bruhat–Tits buildings

For a tame supercuspidal representation π \pi of a connected reductive p p -adic group G G , we establish two distinct and complementary sufficient conditions, formulated in terms of the geometry of the Bruhat–Tits building of G G , for the irreducible components of its restriction to a maximal compact subgroup to occur in a representation of G G which is not inertially equivalent to π \pi . The consequence is a set of broadly applicable tools for addressing the branching rules of π \pi and the unicity of [ G , π ] G [G,\pi ]_G -types.

Equivariant multiplicities of simply-laced type flag minors

Let g \mathfrak {g} be a finite simply-laced type simple Lie algebra. Baumann-Kamnitzer-Knutson recently defined an algebra morphism D ¯ \overline {D} on the coordinate ring C [ N ] \mathbb {C}[N] related to Brion’s equivariant multiplicities via the geometric Satake correspondence. This map is known to take distinguished values on the elements of the MV basis corresponding to smooth MV cycles, as well as on the elements of the dual canonical basis corresponding to Kleshchev-Ram’s strongly homogeneous modules over quiver Hecke algebras. In this paper we show that when g \mathfrak {g} is of type A n A_n or D 4 D_4 , the map D ¯ \overline {D} takes similar distinguished values on the set of all flag minors of C [ N ] \mathbb {C}[N] , raising the question of the smoothness of the corresponding MV cycles. We also exhibit certain relations between the values of D ¯ \overline {D} on flag minors belonging to the same standard seed, and we show that in any A D E ADE type these relations are preserved under cluster mutations from one standard seed to another. The proofs of these results partly rely on Kang-Kashiwara-Kim-Oh’s monoidal categorification of the cluster structure of C [ N ] \mathbb {C}[N] via representations of quiver Hecke algebras.

A proof of Casselman’s comparison theorem

Let G G be a real linear reductive group and K K be a maximal compact subgroup. Let P P be a minimal parabolic subgroup of G G with complexified Lie algebra p \mathfrak {p} , and n \mathfrak {n} be its nilradical. In this paper we show that: for any admissible finitely generated moderate growth smooth Fréchet representation V V of G G , the inclusion V K ⊂ V V_{K}\subset V induces isomorphisms H i ( n , V K ) ≅ H i ( n , V ) H_{i}(\mathfrak {n},V_{K})\cong H_{i}(\mathfrak {n},V) ( i ≥ 0 i\geq 0 ), where V K V_{K} denotes the ( g , K ) (\mathfrak {g},K) module of K K finite vectors in V V . This is called Casselman’s comparison theorem (see Henryk Hecht and Joseph L. Taylor [A remark on Casselman’s comparison theorem, Birkhäuser Boston, Boston, Ma, 1998, pp. 139–146]). As a consequence, we show that: for any k ≥ 1 k\geq 1 , n k V \mathfrak {n}^{k}V is a closed subspace of V V and the inclusion V K ⊂ V V_{K}\subset V induces an isomorphism V K / n k V K = V / n k V V_{K}/\mathfrak {n}^{k}V_{K}= V/\mathfrak {n}^{k}V . This strengthens Casselman’s automatic continuity theorem (see W. Casselman [Canad. J. Math. 41 (1989), pp. 385–438] and Nolan R. Wallach [Real reductive groups, Academic Press, Boston, MA, 1992]).

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.

Quasi-split symmetric pairs of 𝑈(𝔰𝔩_{𝔫}) and Steinberg varieties of classical type

We provide a Lagrangian construction for the fixed-point subalgebra, together with its idempotent form, in a quasi-split symmetric pair of type A n − 1 A_{n-1} . This is obtained inside the limit of a projective system of Borel-Moore homologies of the Steinberg varieties of n n -step isotropic flag varieties. Arising from the construction are a basis of homological origin for the idempotent form and a geometric realization of rational modules.

Splitting fields of real irreducible representations of finite groups

We show that any irreducible representation ρ \rho of a finite group G G of exponent n n , realisable over R \mathbb {R} , is realisable over the field E ≔ Q ( ζ n ) ∩ R E≔\mathbb {Q}(\zeta _n)\cap \mathbb {R} of real cyclotomic numbers of order n n , and describe an algorithmic procedure transforming a realisation of ρ \rho over Q ( ζ n ) \mathbb {Q}(\zeta _n) to one over E E .

Local Langlands correspondence for unitary groups via theta lifts

Using the theta correspondence, we extend the classification of irreducible representations of quasi-split unitary groups (the so-called local Langlands correspondence, which is due to Mok) to non quasi-split unitary groups. We also prove that our classification satisfies some good properties, which characterize it uniquely. In particular, this paper provides an alternative approach to the works of Kaletha-Mínguez-Shin-White and Mœglin-Renard.

Unipotent representations attached to the principal nilpotent orbit

In this paper, we construct and classify the special unipotent representations of a real reductive group attached to the principal nilpotent orbit. We give formulas for the K \mathbf {K} -types, associated varieties, and Langlands parameters of all such representations.

Parabolic induction via the parabolic pro-𝑝 Iwahori–Hecke algebra

Let G \mathbf {G} be a connected reductive group defined over a locally compact non-archimedean field F F , let P \mathbf {P} be a parabolic subgroup with Levi M \mathbf {M} and compatible with a pro- p p Iwahori subgroup of G ≔ G ( F ) G ≔\mathbf {G}(F) . Let R R be a commutative unital ring. We introduce the parabolic pro- p p Iwahori–Hecke R R -algebra H R ( P ) \mathcal {H}_R(P) of P ≔ P ( F ) P ≔\mathbf {P}(F) and construct two R R -algebra morphisms Θ M P : H R ( P ) → H R ( M ) \Theta ^P_M\colon \mathcal {H}_R(P)\to \mathcal {H}_R(M) and Ξ G P : H R ( P ) → H R ( G ) \Xi ^P_G\colon \mathcal {H}_R(P) \to \mathcal {H}_R(G) into the pro- p p Iwahori–Hecke R R -algebra of M ≔ M ( F ) M ≔\mathbf {M}(F) and G G , respectively. We prove that the resulting functor Mod ⁡ - H R ( M ) → Mod ⁡ - H R ( G ) \operatorname {Mod}\text {-}\mathcal {H}_R(M) \to \operatorname {Mod}\text {-}\mathcal {H}_R(G) from the category of right H R ( M ) \mathcal {H}_R(M) -modules to the category of right H R ( G ) \mathcal {H}_R(G) -modules (obtained by pulling back via Θ M P \Theta ^P_M and extension of scalars along Ξ G P \Xi ^P_G ) coincides with the parabolic induction due to Ollivier–Vignéras. The maps Θ M P \Theta ^P_M and Ξ G P \Xi ^P_G factor through a common subalgebra H R ( M , G ) \mathcal {H}_R(M,G) of H R ( G ) \mathcal {H}_R(G) which is very similar to H R ( M ) \mathcal {H}_R(M) . Studying these algebras H R ( M , G ) \mathcal {H}_R(M,G) for varying ( M , G ) (M,G) we prove a transitivity property for tensor products. As an application we give a new proof of the transitivity of parabolic induction.