Sources · the Bruhat decomposition and the Hecke ring

Iwahori–Matsumoto (1965): what the paper proves

The source of the Concept. Its objects are rederived in the notation of these pages, its statements are restated with their numbers, the ones the Concept depends on are proved, and the paper's own wording is quoted for comparison.

Executive summary Context

[IM65] has three sections. §1 is about the group $\Wt$ (their $DW$) acting on the Euclidean space $\Xs\otimes\R$: alcoves, simple transitivity of $\Waff$ (their $D'W$), the length function $\lambda$, the subgroup $\Om$, and the Coxeter presentation. §2 defines the Iwahori subgroup (their $B$) of a Chevalley group over a field with a discrete valuation and proves the decomposition $G=\bigsqcup_{\sigma\in\Wt}B\omega(\sigma)B$ together with the rules for multiplying double cosets. §3 assumes the residue field finite with $q$ elements and determines the Hecke ring $\mathscr H(G,B)$: the index formula $q^{\lambda(\sigma)}$, the quadratic and braid relations, the presentation, and the twisted tensor product with $\Z[\Om]$. The sign character is the last paragraph read for this site.

Table 1. The results of the paper used on this site.
resultstatement in our notationused on
Prop 1.2, Cor 1.8, Cor 1.16$\Waff$ is generated by $\Saff$, acts simply transitively on alcoves, Coxeter presentationGroups §4
Prop 1.10, Lemma 1.5, §1.7$\len$ is word length; $\len(sw)=\len(w)\pm1$; $\Wt=\Waff\rtimes\Om$, $\Om\simeq\Xs/\Qv$Groups §5
Thm 2.5Iwahori factorization $I=U^-(t\cO)T(\cO)U(\cO)$SL₂ I §4
Prop 2.6, §3.1$\ind(s)=q$ for $s\in\Saff$the Concept §2
Prop 2.8, Cor 2.9$IsI\cdot IwI=IswI$ or $IwI\sqcup IswI$ according to $\len(sw)\gtrless\len(w)$the Concept §2
Prop 2.10, Cor 2.19$\sigma$ normalizes $I$ iff $\sigma\in\Om$; $N(I)/I\simeq\Om$the Concept §2
Thm 2.16$\GF=\bigsqcup_{w\in\Wt}IwI$Groups §6
Prop 3.2$\ind(w)=q^{\len(w)}$the Concept §4
Thm 3.3, Prop 3.4, Cor 3.6generation by $T_s,T_\sigma$; quadratic and braid relations; $T_sT_w$ formulasthe Concept §3–4
Thm 3.5, Prop 3.8presentation of $\Haff$; $\cH=\Z[\Om]\widetilde\otimes\Haff$the Concept §3, SL₂ III §3
§3.2, p. 279the sign character $T_s\mapsto-1$, $T_\sigma\mapsto1$the Concept §6

The paper's objects, rederived in our notation Definition

The paper fixes a complex semisimple Lie algebra $\mathfrak g_{\C}$ and a field $K$ with a discrete valuation, and builds the Chevalley group from them. Its objects correspond to ours as follows; the running example $\SL_2$ is in the last column.

Table 2. Dictionary of objects. The paper's symbols appear only here and in the verbatim quotes.
paperheremeaning$\SL_2$
$K$, $\mathfrak O$, $\mathfrak P$, $\pi$, $k=\mathfrak O/\mathfrak P$, $q$$F$, $\cO$, $t\cO$, $t$, $\Fq$, $q$the local field and its residue field$\Fq((t))$
$G$ (adjoint Chevalley group), $G'$ (its commutator subgroup), $G'B$$\GF$ for adjoint $G$; the subgroup $\Waff$-part$G'B=\bigsqcup_{w\in\Waff}IwI$ (Prop 2.20)$\PGL_2(F)\supset\SL_2(F)$-image
$U$$K=G(\cO)$the maximal compact subgroup (Cor 2.17–2.18)$\SL_2(\cO)$
$B$$I$the Iwahori subgroup, generated by $\mathfrak X_{-\alpha,\mathfrak O}$, $\mathfrak X_{\alpha,\mathfrak P}$ ($\alpha>0$) and $\mathfrak H_{\mathfrak O}$lower triangular modulo $t$ (transpose of ours)
$\Wt=DW$, $D'W$, $\Om$$\Wt$, $\Waff$, $\Om$$D$ = translations by $P_r^\perp$, $D'$ by $P^\perp$$\Z\rtimes\Z/2$, same, $1$
$w_0,w_1,\dots,w_l$$s_0,s_1,\dots,s_l$reflections in the walls of the fundamental cell $\mathfrak D_0$$s_0,s_1$
$\lambda(\sigma)$, $l(\sigma)$$\len(w)$hyperplane count; word length (equal by Prop 1.10)number of letters
$\omega(\sigma)$, $B\omega(\sigma)B$$\dot w$, $IwI$a representative and the double coset—
$\mathrm{ind}(x)$$\ind(x)=|IxI/I|$$[B:B\cap x^{-1}Bx]$$q^{\len}$
$\mathscr H(G,B)$, $S_\sigma$, $S_i$$\cH$ over $\Z$, $T_w$, $T_{s_i}$the Hecke ring and its basis$\langle T_{s_0},T_{s_1}\rangle$
$\mathfrak D_0$, $\mathfrak D$$A_0$, alcovesthe fundamental cell $0<(\alpha,x)<1$ (positive chamber: see §5)$(0,1)$ there, $(-1,0)$ here

The statements the Concept depends on, restated and proved Proved (cited)

(decomposition and products of double cosets) [IM65, Thm 2.16, Prop 2.8, Prop 2.6, Prop 2.10] Proved (cited)

For $G$ a Chevalley group over $F$ (adjoint, or its commutator subgroup) and $I$ its Iwahori subgroup:

(a)$\GF=\bigsqcup_{w\in\Wt}IwI$.
(b)For $s\in\Saff$, $w\in\Wt$: $IsI\cdot IwI=IswI$ if $\len(sw)>\len(w)$, and $IsI\cdot IwI=IwI\cup IswI$ if $\len(sw)<\len(w)$.
(c)$I=\bigsqcup_{\nu\in\Fq}\Gamma_s\,x_{-\alpha_s}(\tilde\nu)$ with $\Gamma_s=I\cap\dot s^{-1}I\dot s$ (for $s=s_0$ the root subgroup is $x_{\alpha_0}(t\tilde\nu)$); hence $\ind(s)=q$.
(d)$\dot w$ normalizes $I$ if and only if $w\in\Om$.
The paper's proof, in outline
  1. The Iwahori factorization $B=\mathfrak U_{\mathfrak P}\mathfrak H_{\mathfrak O}\mathfrak V_{\mathfrak O}$ (Thm 2.5) is obtained from the Bruhat decomposition of $U=G(\cO)$ relative to $B$ (Prop 2.4), itself pulled back from the Bruhat decomposition of $G(\Fq)$ along reduction modulo $\pi$.
  2. (c) is Prop 2.6: a rank-one computation in each $\Phi_{\alpha_i}(\SL_2)$, exactly as in SL₂ I, Lemma 4.1.
  3. (b) is Prop 2.8. Part (i) uses the criterion of Prop 1.28–1.29 for $\lambda(w_i\sigma)>\lambda(\sigma)$ to conjugate the root subgroup $\mathfrak X_{-\alpha_i,\mathfrak O}$ into $B$; part (ii) uses (c) and the rank-one identity $x_{-\alpha}(u)\in Bw_iB$ for a unit $u$.
  4. Disjointness in (a) is Prop 2.13, an induction on $\lambda(\sigma)$ using (b); that the union is everything is Thm 2.16, which uses that $B\mathfrak W B$ is a subgroup (Lemma 2.14) containing $B$ and the translations, hence all of $G$ (Lemma 2.15). (d) is Prop 2.10, from the explicit description of $\omega(\sigma)B\omega(\sigma)^{-1}$.

(index formula) [IM65, Prop 3.2] Proved here

$$\ind(w)=q^{\len(w)}\qquad(w\in\Wt).$$
The paper's proof

The paper first shows that $\mathrm{ind}$ extends to a ring homomorphism $\mathscr H(G,B)\to\Z$ (p. 276; our Hecke algebras, Lemma 3.2). For $\sigma=\rho\tau$ with $\rho\in\Om$ and a reduced expression $\tau=w_{i_1}\cdots w_{i_r}$, (b)(i) and Cor 2.9 give $S_\sigma=S_\rho S_{i_1}\cdots S_{i_r}$ (p. 276; this is Thm 3.3). Applying $\mathrm{ind}$: $\mathrm{ind}(S_\sigma)=\mathrm{ind}(S_\rho)\,q^{r}=q^{\lambda(\sigma)}$, since $\mathrm{ind}(S_\rho)=1$ by (d). The step $S_iS_{\tau'}=S_{w_i\tau'}$ when $\lambda(w_i\tau')>\lambda(\tau')$ is where (b)(i) enters; the Concept gives the detailed count behind it (the Concept §4).

(relations) [IM65, Prop 3.4, Cor 3.6] Proved here

(i)$T_s^2=q\,T_e+(q-1)\,T_s$ for $s\in\Saff$.
(ii)Braid relations: $T_sT_t=T_tT_s$, $T_sT_tT_s=T_tT_sT_t$, $(T_sT_t)^2=(T_tT_s)^2$, $(T_sT_t)^3=(T_tT_s)^3$ according as the angle between the walls of $s,t$ is $\pi/2$, $2\pi/3$, $3\pi/4$, $5\pi/6$.
(iii)$T_sT_w=T_{sw}$ if $\len(sw)>\len(w)$; $T_sT_w=qT_{sw}+(q-1)T_w$ if $\len(sw)<\len(w)$; and the same on the right.
The paper's proof of (i)

By (b)(ii) with $w=s$, $IsI\cdot IsI=IsI\cup I$, so $S_i^2=\lambda\cdot1+\mu S_i$ with $\lambda=|B\backslash B\omega(w_i)^{-1}B\cap B\omega(w_i)B|=|B\backslash B\omega(w_i)B|=q$ and $\mu\ge0$. Applying $\mathrm{ind}$: $q^2=\lambda+\mu q$, so $\mu=q-1$ (p. 277). (ii) is deduced from (iii) and the fact that both sides of a braid relation are reduced expressions of the same element (Prop 1.15 and Thm 3.3). (iii) is Cor 3.6, deduced from Thm 3.3 and (i).

(presentation, and the twist by $\Om$) [IM65, Thm 3.5, Prop 3.8] Proved (cited)

Let $\Haff\subset\cH$ be the span of the $T_w$, $w\in\Waff$. The ring homomorphism from the free algebra on symbols $\Delta_s$ ($s\in\Saff$) to $\Haff$, $\Delta_s\mapsto T_s$, is surjective with kernel the ideal generated by the quadratic relations (i) and the braid relations (ii). Moreover $T_\sigma T_sT_\sigma^{-1}=T_{\sigma s\sigma^{-1}}$ for $\sigma\in\Om$, and

$$\cH=\C[\Om]\,\widetilde\otimes\,\Haff,\qquad(\rho\otimes T_w)(\rho'\otimes T_{w'})=\rho\rho'\otimes T_{\rho'^{-1}w\rho'}T_{w'} .$$
The paper's proof, in outline

Surjectivity is Thm 3.3. For the kernel, let $\mathfrak F_0$ be the span of the products $\bar\Delta(\theta(\sigma))=\Delta_{i_1}\cdots\Delta_{i_r}$ along one chosen reduced expression of each $\sigma\in\Waff$ in the quotient by the ideal $\mathfrak a$ of the relations. By Prop 1.15 (Matsumoto's lemma: any two reduced words are connected by braid moves) the product does not depend on the reduced expression, and by Cor 1.11 and Lemma 1.5 the products $\Delta_i\bar\Delta(\theta(\tau))$ lie in $\mathfrak F_0$ in both cases $\lambda(w_i\tau)=\lambda(\tau)\pm1$. So $\mathfrak F_0$ is the whole quotient, which is therefore spanned by $|\Waff|$ elements mapping onto the basis $S_\sigma$: the map is an isomorphism (pp. 277–278). Prop 3.8 then follows from $S_\rho S_\sigma=S_{\rho\sigma}$ and $S_\rho S_\sigma S_\rho^{-1}=S_{\rho\sigma\rho^{-1}}$ (p. 279).

(the sign character) [IM65, §3.2, p. 279] Proved (cited)

$T_s\mapsto-1$ ($s\in\Saff$), $T_\sigma\mapsto1$ ($\sigma\in\Om$) extends uniquely to a ring homomorphism $\eps:\cH\to\Z$.

Proof

By the presentation it suffices to check the relations: $(-1)^2=q+(q-1)(-1)$, both sides of a braid relation become $(-1)^m$, and $\eps(T_\sigma T_sT_\sigma^{-1})=-1=\eps(T_{\sigma s\sigma^{-1}})$. This is the paper's remark; the consequence $\eps(\one_K)=0$ for $W\neq1$ is drawn on the Concept §6.

Dictionary to the paper's conventions, with its wording Context
Verbatim wording (pages of the Numdam scan)
“Theorem 2.16. — $G=B\mathfrak WB=\bigcup_{\sigma\in\widetilde W}B\omega(\sigma)B$ (disjoint union).”[IM65, p. 268]
“Proposition 2.8. — Assume that $\mathfrak g_{\mathbf C}$ is simple. Let $i$ be an integer with $0\le i\le l$ and $\sigma$ an element in $\widetilde W=DW$. Then (i) if $\lambda(w_i\sigma)>\lambda(\sigma)$, we have $B\omega(w_i)B\omega(\sigma)B=B\omega(w_i\sigma)B$; (ii) if $\lambda(w_i\sigma)<\lambda(\sigma)$, we have $B\omega(w_i)B\omega(\sigma)B=B\omega(\sigma)B\cup B\omega(w_i\sigma)B$.”[IM65, p. 265]
“Proposition 3.2. — $\mathrm{ind}(\omega(\sigma))=\mathrm{ind}(S_\sigma)=q^{\lambda(\sigma)}$ for any $\sigma\in DW$.”[IM65, p. 276]
“Proposition 3.4. — (i) $S_i^2=q.1+(q-1)S_i$ for $i=0,\dots,l$. (ii) $S_iS_j=S_jS_i$ if $\theta_{ij}=\pi/2$, $S_iS_jS_i=S_jS_iS_j$ if $\theta_{ij}=2\pi/3$, $(S_iS_j)^2=(S_jS_i)^2$ if $\theta_{ij}=3\pi/4$, $(S_iS_j)^3=(S_jS_i)^3$ if $\theta_{ij}=5\pi/6$.”[IM65, p. 277]
“Theorem 3.5. — Let $\mathfrak F$ be the free ring over $\mathbf Z$ generated by $\Delta_0,\Delta_1,\dots,\Delta_l$ together with the unit element $1$. Let $\varphi$ be the ring homomorphism from $\mathfrak F$ onto $\mathscr H(G'B,B)$ defined by $\varphi(\Delta_i)=S_i$ ($0\le i\le l$). Then the kernel of $\varphi$ coincides with the ideal $\mathfrak a$ of $\mathfrak F$ generated by the following elements: $\Delta_i^2-(q.1+(q-1)\Delta_i)$ ($0\le i\le l$), $\Delta_i\Delta_j-\Delta_j\Delta_i$ (for $\theta_{ij}=\pi/2$), $\Delta_i\Delta_j\Delta_i-\Delta_j\Delta_i\Delta_j$ (for $\theta_{ij}=2\pi/3$), $(\Delta_i\Delta_j)^2-(\Delta_j\Delta_i)^2$ (for $\theta_{ij}=3\pi/4$), $(\Delta_i\Delta_j)^3-(\Delta_j\Delta_i)^3$ (for $\theta_{ij}=5\pi/6$).”[IM65, p. 277]
“Proposition 3.8. — $\mathscr H(G,B)=\mathbf Z[\Omega]\widetilde\otimes_{\mathbf Z}\mathscr H(G'B,B)$.”[IM65, p. 279]
“3.2. As an application of Th. 3.5 and Prop. 3.8, we see that $S_i\to-1$ ($0\le i\le l$), $S_\rho\to1$ ($\rho\in\Omega$) can be extended uniquely to a homomorphism from $\mathscr H(G,B)$ into $\mathbf Z$.”[IM65, p. 279]

What does not transfer to the present setting Context