An affine Gindikin-Karpelevich formula

Iwahori–Hecke algebras for p-adic loop groups

“…It has a vector space basis T x for x ∈ W X where T x is the characteristic function of the double coset I x I, x ∈ W X . In this paper we show the following result, which follows from the finiteness theorems in [3] or [2].…”

“…We conclude by pointing out that a limit of the spherical function (see [2]) may be used to compute the Gindikin-Karpelevic integral. 5 This integral is the local input needed in a generalization of the Langlands-Shahidi method to loop groups-i.e.…”

“…Aside from the Cartan decomposition, this is a quite standard extension of the finite-dimensional case, but we could not find a suitable reference for certain results we needed. Along the way, we realized that a criterion established earlier for detecting when unipotent elements are integral (see [2,Lemma 3.3]) yields a proof of the Cartan decomposition that may extend to a general KacMoody group (the proofs of the Cartan decomposition in [15] and [3] do not seem to extend). We decided to include a sketch of this proof in the affine case in Appendix A, but we do not pursue the general Kac-Moody case here.…”

“…Subsequently, new proofs of this result have been found. One such proof is given in [2] which works in an "elementary" context (essentially using representation theory of affine Lie algebras and structure theory of the p-adic loop group). Note that in loc.…”

“…Here we sketch an alternative construction which uses in an essential way the main finiteness result of [2]. While this work was in preparation, there has appeared yet another approach to proving these finiteness results by Gaussent and Rousseau [16], which works in the setting of general Kac-Moody groups.…”