Bruhat, Cartan and Iwasawa decompositions

Bruhat, Cartan and Iwasawa decompositions

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

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(𝔽) 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

𝔽= ((t)) 𝔬=[[t]] k=, 𝔽=p 𝔬=p k=𝔽p, 𝔽= 𝔽q((t)) 𝔬= 𝔽q[[t]] k=𝔽q,
where p is the field of p-adic numbers, p is the ring of p-adic integers, and 𝔽q is the finite field with q elements. For clarity of presentation we shall work in the first case where 𝔽= ((t)). The diagram
𝔽 | 𝔬 evt=0 k=𝔬/𝔭 gives G = G( ((t)) ) | | K = G( [[t]] ) evt=0 G() | | | I = evt=0 -1(B)) evt=0 B(),
where B() is the "Borel subgroup" of "upper triangular matrices" in G(). The loop group is G=G( ((t)) ), I is the Iwahori subgroup of G,
G()/B() is the   flag variety, G/I is the  affine flag variety, and G/K is the  loop Grassmanian.
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].

Let W0 be the Weyl group of G(), W=W0 𝔥 the affine Weyl group, and U- the subgroup of "unipotent lower triangular" matrices in G(𝔽) and 𝔥+ the set of dominant elements of 𝔥. Then Bruhat decomposition G= wW0 BwB,SPACE K= wW0 IwI, Iwahori decompositionSPA G= wW IwI, G= vW U-vI Cartan decomposition G= λ 𝔥+ KtλK , G= μ 𝔥 U- tμK SPAIwasawa decomposition

The Mirković-Vilonen intersections are IwI U-vI and KtλK U- tμK , for v,wW, λ 𝔥+ and μ 𝔥 .

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 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].


[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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