General Bruhat decomposition (with parabolic not necessarily Borel)

Question

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 ?

Thanks in advance.

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)...

Answer 1

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.

Answer 2

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 $P_\emptyset s_\beta P_\alpha$.