An official website of the United States government
Abstract A spline is an assignment of polynomials to the vertices of a graph whose edges are labeled by ideals, where the difference of two polynomials labeling adjacent vertices must belong to the corresponding ideal. The set of splines forms a ring. We consider spline rings where the underlying graph is the Cayley graph of a symmetric group generated by a collection of transpositions. These rings generalize the GKM construction for equivariant cohomology rings of flag, regular semisimple Hessenberg and permutohedral varieties. These cohomology rings carry two actions of the symmetric group$$S_n$$whose graded characters are both of general interest in algebraic combinatorics. In this paper, we generalize the graded$$S_n$$-representations from the cohomologies of the above varieties to splines on Cayley graphs of$$S_n$$and then (1) give explicit module and ring generators for whenever the$$S_n$$-generating set is minimal, (2) give a combinatorial characterization of when graded pieces of one$$S_n$$-representation is trivial, and (3) compute the first degree piece of both graded characters for all generating sets.
more »
« less
This content will become publicly available on March 7, 2026
Geometric $L$-packets of Howe-unramified toral supercuspidal representations
We show thatL-packets of toral supercuspidal representations arising from unramified maximal tori ofp-adic groups are realized by Deligne–Lusztig varieties for parahoric subgroups. We prove this by exhibiting a direct comparison between the cohomology of these varieties and algebraic constructions of supercuspidal representations. Our approach is to establish that toral irreducible representations are uniquely determined by the values of their characters on a domain of sufficiently regular elements. This is an analogue of Harish-Chandra’s characterization of real discrete series representations by their characters on regular elements of compact maximal tori, a characterization which Langlands relied on in his construction ofL-packets of these representations. In parallel to the real case, we characterize the members of Kaletha’s toralL-packets by their characters on sufficiently regular elements of elliptic maximal tori.
more »
« less
- PAR ID:
- 10610521
- Publisher / Repository:
- Journal of the European Mathematical Society
- Date Published:
- Journal Name:
- Journal of the European Mathematical Society
- Volume:
- 27
- Issue:
- 4
- ISSN:
- 1435-9855
- Page Range / eLocation ID:
- 1465 to 1526
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We give a geometric construction of representations of parahoric subgroups P P of a reductive group G G over a local field which splits over an unramified extension. These representations correspond to characters θ \theta of unramified maximal tori and, when the torus is elliptic, are expected to give rise to supercuspidal representations of G G . We calculate the character of these P P -representations on a special class of regular semisimple elements of G G . Under a certain regularity condition on θ \theta , we prove that the associated P P -representations are irreducible. This generalizes a construction of Lusztig from the hyperspecial case to the setting of an arbitrary parahoric.more » « less
-
We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p-adic group G arise from pairs (S,\theta), where S is a tame elliptic maximal torus of G, and \theta is a character of S satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p-adic groups.more » « less
-
Supercuspidal representations are usually infinite-dimensional, so the size of such a representation cannot be measured by its dimension; the formal degree is a better alternative. Hiraga, Ichino, and Ikeda conjectured a formula for the formal degree of a supercuspidal in terms of its L-parameter only. Our first main result is to compute the formal degrees of the supercuspidal representations constructed by Yu. Our second result, using the first, is to verify that Kaletha’s regular supercuspidal L-packets satisfy the conjecture.more » « less
-
Abstract For a connected reductive groupGover a nonarchimedean local fieldFof positive characteristic, Genestier-Lafforgue and Fargues-Scholze have attached a semisimple parameter$${\mathcal {L}}^{ss}(\pi )$$to each irreducible representation$$\pi $$. Our first result shows that the Genestier-Lafforgue parameter of a tempered$$\pi $$can be uniquely refined to a tempered L-parameter$${\mathcal {L}}(\pi )$$, thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of$${\mathcal {L}}^{ss}(\pi )$$for unramifiedGand supercuspidal$$\pi $$constructed by induction from an open compact (modulo center) subgroup. If$${\mathcal {L}}^{ss}(\pi )$$is pure in an appropriate sense, we show that$${\mathcal {L}}^{ss}(\pi )$$is ramified (unlessGis a torus). If the inducing subgroup is sufficiently small in a precise sense, we show$$\mathcal {L}^{ss}(\pi )$$is wildly ramified. The proofs are via global arguments, involving the construction of Poincaré series with strict control on ramification when the base curve is$${\mathbb {P}}^1$$and a simple application of Deligne’s Weil II.more » « less
