Bruhat, Cartan and Iwasawa decompositions

## 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

 $\begin{array}{lll}𝔽=ℂ\left(\left(t\right)\right)\phantom{\rule{2em}{0ex}}& 𝔬=ℂ\left[\left[t\right]\right]\phantom{\rule{2em}{0ex}}& k=ℂ,\\ 𝔽={ℚ}_{p}& 𝔬={ℤ}_{p}& k={𝔽}_{p},\\ 𝔽={𝔽}_{q}\left(\left(t\right)\right)& 𝔬={𝔽}_{q}\left[\left[t\right]\right]& k={𝔽}_{q},\end{array}$
where ${ℚ}_{p}$ is the field of $p$-adic numbers, ${ℤ}_{p}$ 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 $𝔽=ℂ\left(\left(t\right)\right)$. The diagram
 $\begin{array}{ccc}𝔽& & \\ \cup |& & \\ 𝔬& \stackrel{{\mathrm{ev}}_{t=0}}{⟶}& k=𝔬/𝔭\end{array}\phantom{\rule{3em}{0ex}}$ gives $\phantom{\rule{3em}{0ex}}\begin{array}{ccccc}G& =& G\left(ℂ\left(\left(t\right)\right)\right)& & \\ \cup |& & \cup |& & \\ K& =& G\left(ℂ\left[\left[t\right]\right]\right)& \stackrel{{\mathrm{ev}}_{t=0}}{⟶}& G\left(ℂ\right)\\ \cup |& & \cup |& & \cup |\\ I& =& {{\mathrm{ev}}_{t=0}}^{-1}\left(Bℂ\right)\right)& \stackrel{{\mathrm{ev}}_{t=0}}{⟶}& B\left(ℂ\right),\end{array}$
where $B\left(ℂ\right)$ is the "Borel subgroup" of "upper triangular matrices" in $G\left(ℂ\right)$. The loop group is $G=G\left(ℂ\left(\left(t\right)\right)\right)$, $I$ is the Iwahori subgroup of $G$, 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 ${W}_{0}$ be the Weyl group of $G\left(ℂ\right)$, $W={W}_{0}⋉{𝔥}_{ℤ}$ the affine Weyl group, and ${U}^{-}$ the subgroup of "unipotent lower triangular" matrices in $G\left(𝔽\right)$ and ${𝔥}_{ℤ}^{+}$ the set of dominant elements of ${𝔥}_{ℤ}$. Then $Bruhat decomposition G=⨆ w∈W0 BwB,SPACE K= ⨆ w∈W0 IwI, Iwahori decompositionSPA G= ⨆ w∈W IwI, G= ⨆v∈W U-vI Cartan decomposition G=⨆ λ∨ ∈𝔥ℤ+ Ktλ∨K , G=⨆ μ∨∈ 𝔥ℤ U- tμ∨K SPAIwasawa decomposition$

The Mirković-Vilonen intersections are $IwI ∩ U-vI and Ktλ∨K ∩ U- tμ∨K ,$ for $v,w\in W$, ${\lambda }^{\vee }\in {𝔥}_{ℤ}^{+}$ and ${\mu }^{\vee }\in {𝔥}_{ℤ}$.

## Notes and References

We have, intentionally, not given precise definitions of the objects in Theorem 1.1. Even in the classical case, the definition of $𝔥ℤ$ 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??????