A proof of the linear Arithmetic Fundamental Lemma for

2020 ◽  
pp. 1-47
Author(s):  
Qirui Li
Keyword(s):  
Local Field ◽  
Intersection Number ◽  
Hecke Algebra ◽  
Quadratic Extension ◽  
Unit Element ◽  
Linear Arithmetic ◽  
Fundamental Lemma ◽  
Special Cases

Abstract Let $K/F$ be an unramified quadratic extension of a non-Archimedean local field. In a previous work [1], we proved a formula for the intersection number on Lubin–Tate spaces. The main result of this article is an algorithm for computation of this formula in certain special cases. As an application, we prove the linear Arithmetic Fundamental Lemma for $ \operatorname {{\mathrm {GL}}}_4$ with the unit element in the spherical Hecke Algebra.

scholarly journals Conjectures and results about parabolic induction of representations of $${\text {GL}}_n(F)$$

2020 ◽  
Vol 222 (3) ◽  
pp. 695-747
Author(s):  
Erez Lapid ◽  
Alberto Mínguez
Keyword(s):  
Linear Group ◽  
Local Field ◽  
General Linear Group ◽  
Irreducible Representations ◽  
General Linear ◽  
Parabolic Induction ◽  
Geometric Conditions ◽  
Irreducible Components ◽  
Special Cases

Abstract In 1980 Zelevinsky introduced certain commuting varieties whose irreducible components classify complex, irreducible representations of the general linear group over a non-archimedean local field with a given supercuspidal support. We formulate geometric conditions for certain triples of such components and conjecture that these conditions are related to irreducibility of parabolic induction. The conditions are in the spirit of the Geiss–Leclerc–Schröer condition that occurs in the conjectural characterization of $$\square $$ □ -irreducible representations. We verify some special cases of the new conjecture and check that the geometric and representation-theoretic conditions are compatible in various ways.

scholarly journals Character Polynomials, their q-Analogs and the Kronecker Product

10.37236/85 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
A. M. Garsia ◽  
A. Goupil
Keyword(s):  
Numerical Calculation ◽  
Kronecker Product ◽  
Hecke Algebra ◽  
Kronecker Products ◽  
Proper Role ◽  
Special Cases ◽  
Natural Counterpart ◽  
Kronecker Coefficients ◽  
Combinatorial Construction

The numerical calculation of character values as well as Kronecker coefficients can efficently be carried out by means of character polynomials. Yet these polynomials do not seem to have been given a proper role in present day literature or software. To show their remarkable simplicity we give here an "umbral" version and a recursive combinatorial construction. We also show that these polynomials have a natural counterpart in the standard Hecke algebra ${\cal H}_n(q\, )$. Their relation to Kronecker products is brought to the fore, as well as special cases and applications. This paper may also be used as a tutorial for working with character polynomials in the computation of Kronecker coefficients.

scholarly journals HECKE ALGEBRAS FOR INNER FORMS OF -ADIC SPECIAL LINEAR GROUPS

2015 ◽  
Vol 16 (2) ◽  
pp. 351-419 ◽  
Author(s):  
Anne-Marie Aubert ◽  
Paul Baum ◽  
Roger Plymen ◽  
Maarten Solleveld
Keyword(s):  
Finite Group ◽  
Local Field ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Linear Groups ◽  
Special Linear Groups ◽  
Affine Hecke Algebra ◽  
Morita Equivalent ◽  
Affine Hecke Algebras ◽  
A Finite Group

Let$F$be a non-Archimedean local field, and let$G^{\sharp }$be the group of$F$-rational points of an inner form of$\text{SL}_{n}$. We study Hecke algebras for all Bernstein components of$G^{\sharp }$, via restriction from an inner form$G$of$\text{GL}_{n}(F)$.For any packet of L-indistinguishable Bernstein components, we exhibit an explicit algebra whose module category is equivalent to the associated category of complex smooth$G^{\sharp }$-representations. This algebra comes from an idempotent in the full Hecke algebra of$G^{\sharp }$, and the idempotent is derived from a type for$G$. We show that the Hecke algebras for Bernstein components of$G^{\sharp }$are similar to affine Hecke algebras of type$A$, yet in many cases are not Morita equivalent to any crossed product of an affine Hecke algebra with a finite group.

scholarly journals Application of the Bruhat-Tits tree of SU3(h) to some Ã2 groups

1998 ◽  
Vol 64 (3) ◽  
pp. 329-344 ◽  
Author(s):  
Donald I. Cartwright ◽  
Tim Steger
Keyword(s):  
Local Field ◽  
Quadratic Extension ◽  
Tits Building

AbstractLet K be a nonarchimedean local field, let L be a separable quadratic extension of K, and let h denote a nondegenerate sesquilinear formk on L3. The Bruhat-Tits building associated with SU3(h) is a tree. This is applied to the study of certain groups acting simply transitively on vertices of the building associated with SL(3, F), F = Q3 or F3((X)).

scholarly journals Arithmetic intersection on GSpin Rapoport–Zink spaces

2018 ◽  
Vol 154 (7) ◽  
pp. 1407-1440 ◽  
Author(s):  
Chao Li ◽  
Yihang Zhu
Keyword(s):  
Explicit Formula ◽  
Intersection Number ◽  
Shimura Varieties ◽  
Local Problem ◽  
Fundamental Lemma

We prove an explicit formula for the arithmetic intersection number of diagonal cycles on GSpin Rapoport–Zink spaces in the minuscule case. This is a local problem arising from the arithmetic Gan–Gross–Prasad conjecture for orthogonal Shimura varieties. Our formula can be viewed as an orthogonal counterpart of the arithmetic–geometric side of the arithmetic fundamental lemma proved by Rapoport–Terstiege–Zhang in the minuscule case.

scholarly journals Epsilon factors of symplectic type characters in the wild case

2021 ◽  
Vol 33 (2) ◽  
pp. 569-577
Author(s):  
Sazzad Ali Biswas
Keyword(s):  
Local Field ◽  
Characteristic Zero ◽  
Quadratic Extension ◽  
Multiplicative Character ◽  
In The Wild ◽  
Epsilon Factors

Abstract By work of John Tate we can associate an epsilon factor with every multiplicative character of a local field. In this paper, we determine the explicit signs of the epsilon factors for symplectic type characters of K × {K^{\times}} , where K / F {K/F} is a wildly ramified quadratic extension of a non-Archimedean local field F of characteristic zero.

scholarly journals Evaluations of Hecke Algebra Traces at Kazhdan-Lusztig Basis Elements

10.37236/5021 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Samuel Clearman ◽  
Matthew Hyatt ◽  
Brittany Shelton ◽  
Mark Skandera
Keyword(s):  
Hecke Algebra ◽  
Combinatorial Interpretation ◽  
Quasisymmetric Functions ◽  
Irreducible Characters ◽  
Algebraic Interpretation ◽  
Special Cases

For irreducible characters $\{ \chi_q^\lambda \,|\, \lambda \vdash n \}$, induced sign characters $\{ \epsilon_q^\lambda \,|\, \lambda \vdash n \}$, and induced trivial characters $\{ \eta_q^\lambda \,|\, \lambda \vdash n \}$ of the Hecke algebra $H_n(q)$, and Kazhdan-Lusztig basis elements $C'_w(q)$ with $w$ avoiding the patterns 3412 and 4231, we combinatorially interpret the polynomials $\smash{\chi_q^\lambda(q^{\ell(w)/2} C'_w(q))}$, $\smash{\epsilon_q^\lambda(q^{\ell(w)/2} C'_w(q))}$, and $\smash{\eta_q^\lambda(q^{\ell(w)/2} C'_w(q))}$. This gives a new algebraic interpretation of chromatic quasisymmetric functions of Shareshian and Wachs, and a new combinatorial interpretation of special cases of results of Haiman. We prove similar results for other $H_n(q)$-traces, and confirm a formula conjectured by Haiman.

scholarly journals The remaining cases of the Kramer–Tunnell conjecture

2016 ◽  
Vol 152 (11) ◽  
pp. 2255-2268
Author(s):  
Kęstutis Česnavičius ◽  
Naoki Imai
Keyword(s):  
Elliptic Curve ◽  
Local Field ◽  
Positive Characteristic ◽  
Quadratic Extension ◽  
Base Change ◽  
Local Fields ◽  
Root Number ◽  
Characteristic Case ◽  
Local Root

For an elliptic curve $E$ over a local field $K$ and a separable quadratic extension of $K$, motivated by connections to the Birch and Swinnerton-Dyer conjecture, Kramer and Tunnell have conjectured a formula for computing the local root number of the base change of $E$ to the quadratic extension in terms of a certain norm index. The formula is known in all cases except some where $K$ is of characteristic $2$, and we complete its proof by reducing the positive characteristic case to characteristic $0$. For this reduction, we exploit the principle that local fields of characteristic $p$ can be approximated by finite extensions of $\mathbb{Q}_{p}$: we find an elliptic curve $E^{\prime }$ defined over a $p$-adic field such that all the terms in the Kramer–Tunnell formula for $E^{\prime }$ are equal to those for $E$.

Orbital Integrals on Forms of SL(3), II

1989 ◽  
Vol 41 (3) ◽  
pp. 480-507 ◽  
Author(s):  
R. P. Langlands ◽  
D. Shelstad
Keyword(s):  
Asymptotic Behavior ◽  
Local Field ◽  
Unsolved Problem ◽  
Reductive Group ◽  
Local Problem ◽  
Orbital Integrals ◽  
Special Cases ◽  
Endoscopic Group

In the paper [6] we described in a precise fashion the notion of transfer of orbital integrals from a reductive group over a local field to an endoscopic group. We did not, however, prove the existence of the transfer. This remains, indeed, an unsolved problem, although in [7] we have reduced it to a local problem at the identity.In the present paper we solve this local problem for two special cases, the group SL(3), which is not so interesting, and the group SU(3), and then conclude that transfer exists for any group of type A2.The methods are those of [4], and are based on techniques of Igusa for the study of the asymptotic behavior of integrals on p-adic manifolds. (As observed in [7], the existence of the transfer over archimedean fields is a result of earlier work by Shelstad.)

scholarly journals On the arithmetic fundamental lemma in the minuscule case

2013 ◽  
Vol 149 (10) ◽  
pp. 1631-1666 ◽  
Author(s):  
Michael Rapoport ◽  
Ulrich Terstiege ◽  
Wei Zhang
Keyword(s):  
Symmetric Space ◽  
Trace Formula ◽  
Moduli Space ◽  
Intersection Number ◽  
Fundamental Lemma ◽  
The Third ◽  
Orbital Integral ◽  
Relative Trace Formula ◽  
Divisible Groups

AbstractThe arithmetic fundamental lemma conjecture of the third author connects the derivative of an orbital integral on a symmetric space with an intersection number on a formal moduli space of $p$-divisible groups of Picard type. It arises in the relative trace formula approach to the arithmetic Gan–Gross–Prasad conjecture. We prove this conjecture in the minuscule case.

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