Bruhat, Cartan and Iwasawa decompositions

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 17 January 2012

Bruhat, Cartan and Iwasawa decompositions

A Chevalley group is a group in which row reduction works. This means that it is a group with a special set of generators (the "elementary matrices") and relations which are generalizations of the usual row reduction operations. One way to efficiently encode these generators and relations is with a Kac-Moody Lie algebra $𝔤$. From the data of the Kac-Moody Lie algebra and a choice of a commutative ring or field $𝔽$ the group $G\left(𝔽\right)$ is built by generators and relations following chevalley-Steinberg-Tits.

Of particular interest is the case where $𝔽$ is the field of fractions of $𝔬$, the discrete valuation ring $𝔬$ is the ring of integers in $𝔽$, $𝔭$ is the unique maximal ideal in $𝔬$ and $k=𝔬/𝔭$ is the residue field. The favourite examples are

$\begin{array}{lll}𝔽=ℂ\left(\right(t\left)\right)& 𝔬=ℂ\left[\right[t\left]\right]& k=ℂ,\\ 𝔽={\mathbb{Q}}_p& 𝔬={\mathbb{Z}}_p& k={𝔽}_p,\\ 𝔽={𝔽}_q\left(\right(t\left)\right)& 𝔬={𝔽}_q\left[\right[t\left]\right]& k={𝔽}_q,\end{array}$

where ${\mathbb{Q}}_p$ is the field of $p$-adic numbers, ${\mathbb{Z}}_p$ is the ring of $p$-adic integers, and ${𝔽}_q$ is the finite field with $q$ elements. For clarity of presentation we shall work in the first case where $𝔽=ℂ\left(\right(t\left)\right)$. The diagram

$\begin{array}{ccc}𝔽& & \\ \cup |& & \\ 𝔬& \stackrel{{\mathrm{ev}}_{t=0}}{⟶}& k=𝔬/𝔭\end{array}$ gives $\begin{array}{ccccc}G& =& G(ℂ(\left(t\right)\left)\right)& & \\ \cup |& & \cup |& & \\ K& =& G(ℂ[\left[t\right]\left]\right)& \stackrel{{\mathrm{ev}}_{t=0}}{⟶}& G\left(ℂ\right)\\ \cup |& & \cup |& & \cup |\\ I& =& {{\mathrm{ev}}_{t=0}}^{-1}(Bℂ))& \stackrel{{\mathrm{ev}}_{t=0}}{⟶}& B\left(ℂ\right),\end{array}$

where $B\left(ℂ\right)$ is the "Borel subgroup" of "upper triangular matrices" in $G\left(ℂ\right)$. The loop group is $G=G\left(ℂ\right(\left(t\right)\left)\right)$, $I$ is the Iwahori subgroup of $G$,

$$\begin{array}{ll}G\left(ℂ\right)/B\left(ℂ\right)& \text{is the <strong>flag variety</strong>,}\\ G/I& \text{is the <strong>affine flag variety</strong>, and}\\ G/K& \text{is the <strong>loop Grassmanian</strong>.}\end{array}$$

The primary tool for the study of these varieties (ind-schemes) are the following "classical" double coset decompositions, see [St, Ch.8] and [Mac1, 2.6].

The Mirković-Vilonen intersections are $$IwI\cap U^{-}vI\text{and}K{t}_{{\lambda }^{\vee }}K\cap U^{-}{t}_{{\mu }^{\vee }}K,$$ for $v,w\in W$, ${\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}^{+}$ and ${\mu }^{\vee }\in {𝔥}_{\mathbb{Z}}$.

Notes and References

This page is a retyped version of [PRS, §1].

We have, intentionally, not given precise definitions of the objects in Theorem 1.1. Even in the classical case, the definition of $𝔥\mathbb{Z}$ in Theorem 1.1 is sensitive to small chances in the definition of $G$ (center, completions, etc.) and there are subtleties in making these definitions correctly in general. These issues are partly treated in [Ga1, Theorem 14.10, Lemma 6.14], [GR, Remark 6.10] and [BF, Proposition 3.7].

References

[BF] A. Braverman and M. Finkelberg, Pursuing the double affine Grassmannian I:Transversal slices via instantons on $Ak$-singularities, arXiv:0711.2083. MR??????

[Ga1] H. Garland, Arithmetic theory of loop groups, Publ. Math. Inst. Hautes Études Sci. 52 (1980), 181-312. MR??????

[GR] S. Gaussent and G. Rousseau, Kac-Moody groups, hovels and Littelmann's paths, arXiv:math.GR/0703639. MR??????

[Mac1] I.G. Macdonald, Spherical functions on a group of p-adic type, Publ. Ramanujan Institute No. 2, Madras, 1971. MR??????

[PRS] J. Parkinson, A. Ram and C. Schwer, Combinatorics in affine flag varieties, J. Algebra 321 (2009), 3469-3493. MR??????

[St] R. Steinberg, Lecture notes on Chevalley groups, Yale University, 1967. MR??????

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