# General Bruhat decomposition (with parabolic not necessarily Borel)

Here is the general Bruhat decomposition (which I have seen in various paper but never with a proof or a complete reference).

Let $G$ be a split reductive group, $T$ a split maximal torus and $B$ a Borel subgroup of $G$.

Let $R^+ \subset R$ be the positive roots corresponding to $B$ and $S \subset R^+$ the simple roots of $R^+$. Let $I \subset S$ and $P_I$ the standard parabolic subgroup of $G$ corresponding to $I$.

Finally let $W$ be the Weyl group of $(G,T)$ and $W_I$ the subgroup of $W$ generated by the reflections $(s_\alpha)_{\alpha \in I}$.

Then the general Bruhat decomposition is $$G = \coprod_{W_I \backslash W / W_I} P_I w P_I$$ and $P_I \backslash P_I w P_I$ is an affine variety of dimension $\ell(w)$ where $w$ is of minimal length in the double coset $P_I w P_I$.

My question is : is there a good choice of representatives for $P_I \backslash P_I w P_I$ ? More precisely, I am looking for an analogue of the following bijection (in the case $P=B$ Borel) : $$B \times \lbrace w \rbrace \times U_{w^{-1}} \overset{\sim}{\longrightarrow} BwB$$ where $U$ is the unipotent radical of $B$, $U^-$ its opposite and $U_{w^{-1}}$ is the subgroup $(w^{-1}U^-w) \cap U$. What subgroup of $P_I$ would replace $U_{w^{-1}}$ ?

Also what reference exists for all this ?

Edit : in this course of Casselman I found the following isomorphism of variety (see on top of page 12)

$$P_I \times \lbrace w \rbrace \times \prod_{\alpha \in R^+ \backslash R_I^+ ~|~ w^{-1} \alpha \notin R^+ \backslash R_I^+} N_\alpha \overset{\sim}{\longrightarrow} P_IwP_I$$

with $w \in W$ of minimal length in $W_I \backslash W / W_I$. However this seems not to work with $\mathrm{GL_3}$ : we note $S = \lbrace \alpha, \beta \rbrace$ ; if $I= \lbrace \alpha \rbrace$, $P_I = \left( \begin{smallmatrix} * & * & * \newline * & * & * \newline & & * \end{smallmatrix} \right)$ ; with $w = s_\beta$ the above product is on the set $\lbrace \beta, \alpha + \beta \rbrace$, so the isomorphism should be $P_I s_\beta P_I \cong P_I \times \lbrace s_\beta \rbrace \times \left( \begin{smallmatrix} 1 & 0 & * \newline 0 & 1 & * \newline 0 & 0 & 1 \end{smallmatrix} \right)$, which is false (the element $s_\beta \left( \begin{smallmatrix} 1 & 0 & 0 \newline 1 & 1 & 0 \newline 0 & 0 & 1 \end{smallmatrix} \right)$ is in the left side, not in the right side)...

• The "bijection" you mention near the end is not just a bijection, but an isomorphism of varieties (an important distinction). That being said, have you looked at 21.29(ii),(iv) in the 2nd edition of Borel's "Linear algebraic groups"? Commented Jan 15, 2013 at 15:08
• The canonical references are Springer, Waterhouse, Humpreys, and Borel, all with some algebraic groups in the title. Commented Jan 15, 2013 at 15:28
• @Marc: Does Waterhouse's book discuss the structure of reductive (or just semisimple) groups? I thought it only focuses on general features of affine group schemes, not serious theorems about semisimple groups. Commented Jan 15, 2013 at 15:32
• @kreck: you are right, there is nothing about the specific question at hand. Commented Jan 15, 2013 at 15:37
• Thank you both. I have looked in Borel's 21.29 but I am not sure how to deduce the decomposition I want from theses results. In Springer and Humphreys'book I haven't found the answer either. The only reference that mention such a decomposition is Casselman (see my edit above) but it seems to be false... Commented Jan 16, 2013 at 14:04

EDIT: My comments were too hasty and are being deleted. Looking at the original sources gets a bit confusing due to the generality, so I'm still looking for a more straightforward later expostion in the split case only. (However, most applications tend to involve fields of definition relative to which $G$ is not split.)

The basic Bruhat decomposition (in refined form) expresses the flag variety $G/B$ as a disjoint union over $W$ of Bruhat cells: the cell indexed by $w$ has dimension $\ell(w)$ and is expressed in terms of a product over this many root groups. In the corresponding Tits system there are standard parabolic subgroups containing $B$, so it's natural to investigate the "partial" flag variety $G/P$ by projecting the flag variety onto it and seeing where the Bruhat cells go. This is essentially what the computations in the sources mentioned are getting at. In the split situation, $C(w)$ indeed means $BwB$, and the image of the Bruhat cell in $G/P$ then has the format indicated by Borel at the end of his section 21. Here you use a smallest length representative of a Weyl group element relative in the quotient $W/W_J$ if $J$ defines the parabolic. For instance, when $G= \mathrm{SL}_3$ and $J$ contains one simple reflection, you get a cell decomposition of $G/P$ into three cells of dimension $0,1,2$.

Concerning references, kreck points out the treatment in Borel's second edition, which is partly drawn from the earlier joint work with Tits on reductive groups over arbitrary fields: see especially section 3 of their "complements" paper in Publ. Math. IHES (1972) here.

When looking at these sources, keep in mind that they were motivated especially by the behavior of non-split reductive groups over non-algebraically closed fields; so their statements get technical. In any case, the structure of each cell in $G/P$ is laid out explicitly in the manner of their treatment of double cosets relative to $B$. Naturally there is a choice of Weyl group representatives involved, but otherwise it's much the same as the usual Bruhat cell decomposition.

• Thank you for your answer and the useful reference. The results in Borel & Tits "Compléments à l'article "Groupes réductifs"" seems to apply for double cosets of the type $BwP$ where $B$ is Borel and $P$ standard parabolic (especially Propostion 3.16 where they study the projection of $BwB$ on $G/P$). I see how to deduce the general Bruhat decomposition as above, but not how to get a "good" system of representatives of $P \backslash PwP$. Commented Jan 15, 2013 at 22:56
• In B-T the notation $C(w)$ seems to be used for the double coset $PwP$ with $P$ minimal parabolic (3.1). I don't see clearly the structure of $PwP$ when $P$ is no more minimal. Is there a simple description of the root groups appearing in $P \backslash PwP$ ? Commented Jan 16, 2013 at 0:27
• Thanks again for your help. I am still having issue to deduce from this study how to express $PwP$ as $Pw \prod N_\alpha$ where $\prod N_\alpha$ is the product of some root groups. I am not even sure such decomposition exists if $P$ is not Borel. Commented Jan 16, 2013 at 13:58
• @Arkandias: I don't recognize this type of decomposition of the double coset $PwP$. Note that $G$ doesn't usually have a Tits system with a given parabolic $P$ playing the role of $B$, so the terminology gets tricky when you discuss "general Bruhat decomposition". It might help to consult Casselman directly about his conventions. Commented Jan 16, 2013 at 15:01

As @JimHumphreys has pointed out, [BT2] Borel and Tits - Compléments à l'article: «Groupes réductifs», specifically Proposition 3.16(i, iv), gives the decomposition $$G/P_I = \bigsqcup_{w \in [W_I\backslash W/W_I]} P_I w P_I/P_I$$, where $$[\cdot]$$ denotes the minimal-length double-coset representatives. However, it is false that $$P_I w P_I/P_I$$ is an affine space.

I asked Josh Lansky about this, and he pointed me to Theorem 5.2 of his paper [La] Decomposition of double cosets in $$\mathfrak p$$-adic groups, which (while considering a more general situation involving parahorics instead of parabolics) suggests that, instead of trying directly to describe $$P_I w P_I/P_I$$, instead we should consider $$P_I w P_I/P_I = \bigsqcup_{w' \in [W_I/W_I \cap w W_I w^{-1}]} P_\emptyset w'w P_I/P_I.$$ Then, as you'd expect, each $$P_\emptyset w' w P_I/P_I$$ is an affine space; here you can use [La, Theorem 4.6] and [BT2, Proposition 3.16(ii)], which show that $$P_\emptyset w'w P_\emptyset/P_\emptyset \to P_\emptyset w'w P_I/P_I$$ is an isomorphism of varieties (from an affine space).

Thus, the problematic case of $$\operatorname{GL}_3$$ that you mention now becomes $$P_\alpha s_\beta P_\alpha/P_\alpha = P_\emptyset s_\beta P_\alpha/P_\alpha \sqcup P_\emptyset s_\alpha s_\beta P_\alpha/P_\alpha,$$ where $$P_\emptyset s_\beta P_\alpha/P_\alpha$$ and $$P_\emptyset s_\alpha s_\beta P_\alpha/P_\alpha$$ are $$1$$- and $$2$$-dimensional affine spaces, respectively; and, of course, the element $$s_\beta\begin{pmatrix} 1 \\ 1 & 1 \\ && 1 \end{pmatrix}$$ that you mention lies in $$P_\emptyset s_\beta P_\alpha$$.

• Yes, it is my impression that $B\backslash G/P$ has "canonical" representatives for $B$ the Borel (or Iwahoric), but not in general. I had thought for years that Casselman's approach should work, but it seems not, disappointingly. Commented May 23, 2022 at 18:29
• @paulgarrett, I think that perhaps one should think of $B\backslash G/P$ rather as $U\backslash G/P$ (their equality being a consequence of the Bruhat decomposition). Then the generalisation is to think of $U_I\backslash G/P_J$, which is parameterised by $\operatorname{Trans}(A_J, M_I)/M_J$; and now each $U_I n P_J/P_J$ is affine. Commented May 26, 2022 at 17:10