‎This paper gives several examples of the basic functions‎ ‎introduced in recent years by Ng^o‎. ‎These are mainly‎ ‎conjectures based on computer experiment‎.  

We construct the Plancherel measure corresponding to the Bessel model on the split special orthogonal group of odd degree ≥ 5 and the Whittaker model on a connected split reductive group in general. As an application we prove the inversion formula which expresses the related integral transform in terms of the Satake transform.

In this paper, we present an expository treatment of the Satake transform. This gives an isomorphism between the spherical Hecke algebra of a split reductive group G over a local field and the representation ring of the dual group Ĝ. If one wants to use the Satake isomorphism to convert information on eigenvalues for the Hecke algebra to local L-functions, it has to be made quite explicit. This...

Abstract We prove the Casselman–Shalika formula for unramified groups over a non-archimedean local field by studying action of spherical Hecke algebra on space compact Whittaker functions via twisted Satake transform. This method provides conceptual explanation appearance characters dual group in formula.

The connections between the objects mentioned in the title are used to give a short proof of the Cartan–Helgason theorem and a natural construction of the compactifications.

Last updated: December 10, 2013. Topics: Motivation for the study of spherical representations; Satake isomorphism stated for the general case of a connected reductive group (taking Bruhat-Tits theory as a black box); interpretation (spherical principal series, Satake parameter, representations of dual reductive group); Satake made more explicit for the split case (key calculation); idea of pro...

Kuga-Satake varieties are abelian varieties associated to certain weight two Hodge structures, for example the second cohomology group of a K3 surface. We start with an introduction to Hodge structures and we give a detailed account of the construction of Kuga-Satake varieties. The Hodge conjecture is discussed in section 2. An excellent survey of the Hodge conjecture for abelian varieties is [...

This is a companion paper of arXiv:1909.11492. We prove an equivalence relating representations degenerate orthosymplectic supergroup with the category $SO(N-1,{\mathbb C}[\![t]\!])$-equivariant perverse sheaves on affine Grassmannian $SO_N$. explain how this fits into more general framework conjectures due to Gaiotto and Ben-Zvi, Sakellaridis Venkatesh.

Let F be a finite extension of Qp. Using the mod p Satake transform, we define what it means for an irreducible admissible smooth representation of an F -split p-adic reductive group over Fp to be supersingular. We then give the classification of irreducible admissible smooth GLn(F )-representations over Fp in terms of supersingular representations. As a consequence we deduce that supersingular...

Reproduction Akiko Satake*, Gen Sakurai and Toshinori Kinoshita* Faculty of Environmental Earth Science, Hokkaido University, N10W5, Kita-ku, Sapporo, 060-0810 Japan Ecosystem Informatics Division, National Institute for Agro-Environmental Sciences, 3-1-3 Kannondai, Tsukuba, 305-8604 Japan Division of Biological Science, Graduate School of Science, Nagoya University, Chikusa, Nagoya, 464-8602 J...