Sources · the modern account, the Bernstein presentation and the spherical corner

Haines–Kottwitz–Prasad (2010): the second presentation and the spherical algebra

The source for everything on this site that goes beyond Iwahori–Matsumoto: the form of the relations used by the Concept, the Bernstein presentation and the centre, and the identification of the spherical Hecke algebra with the corner $\eK\cH\eK$ that the proof workspace needs.

Executive summary Context

[HKP10] proves, for a split reductive group over a $p$-adic field, Bernstein's presentation of $\cH$, the description of its centre, Macdonald's formula, the Casselman–Shalika formula and the Lusztig–Kato formula. The method is to study the module $M=\Cc(T(\cO)U(F)\backslash\GF/I)$, show it is free of rank one over $\cH$, and read the structure of $\cH$ off intertwining operators on $M$. Only §1, §2, §4 and §7 are used here.

Table 1. The results of the paper used on this site (arXiv v3 numbering and pages).
result	statement in our notation	used on
§1.2, p. 2	$\cH=\Cc(I\backslash\GF/I)$ with $\vol(I)=1$; basis $T_x=\one_{IxI}$, $x\in\Wt$	the Concept §3
§1.1, §7.1	$\Wt=\Xs\rtimes W=\Waff\rtimes\Om$; $S_{\mathrm{aff}}$ with $s_0=t_{-\tilde\alpha^\vee}s_{\tilde\alpha}$; $\Om$ of length zero	Groups §3–4
footnote 2, p. 3	the alcove of $I$ lies in the negative chamber	Groups §4
§7.2, p. 20 (display)	the Iwahori–Matsumoto relations $T_xT_s$, $T_xT_\sigma$	the Concept §3
proof of Thm 5.6.1, p. 15	$[K:I]=W(q)=\sum_{w\in W}q^{\len(w)}$	Groups §2
§1.7, Lemma 1.7.1, Rmk 1.7.2	$\Hzero\subset\cH$; $\C[\Xs]\otimes\Hzero\simeq\cH$; the elements $\Th_\lambda$	§3
(1.15.2)	the Bernstein relation	§3
Lemma 2.3.1	$\Zc=\C[\Xs]^W$	§3
§4.1, §4.3, (4.6.1)	$\HK=\eK\cH\eK$; Satake $\HK\simeq\C[\Xs]^W$; $h=\eK z$	§4

The paper's objects, rederived in our notation Definition

Table 2. Dictionary of objects, with the instance for $\SL_2$.
paper	here	meaning	$\SL_2$
$F$, $\cO$, $P=(\pi)$, $k$, $q$	$F$, $\cO$, $t\cO$, $\Fq$, $q$	a $p$-adic field there; $\Fq((t))$ here	—
$G\supset B=AN\supset A$; $\bar B=A\bar N$	$G\supset B=TU\supset T$; the opposite Borel	split, defined over $\cO$	upper triangular, diagonal
$K=G(\cO)$, $I$	$K$, $I$	same definitions	$c\in t\cO$
$\widetilde W=N_{G(F)}(A)/A_{\cO}$, $\pi^\mu$, $t_\mu$	$\Wt$, $t^{\mu}$	$\mu\in X_*(A)=\Xs$	$\Z\rtimes\Z/2$
$T_x=1_{IxI}$, $H$	$T_w$, $\cH$	same, with $I$ of measure $1$	—
$H_0=C(I\backslash K/I)$	$\Hzero$	the finite Hecke algebra	$\C T_e\oplus\C T_{s_1}$
$R=\C[X_*(A)]$, $\Theta_\lambda$	$\C[\Xs]$, $\Th_\lambda$	the commutative subalgebra of translations	$\C[\Th_{\alpha^\vee}^{\pm1}]$
$M=C_c(A_{\cO}N\backslash G/I)$, $v_x$	not used elsewhere on this site	the universal unramified principal series, a right $H$-module	—
$e_K=1_K/\mathrm{meas}(K)$, $H_K$	$\eK$, $\HK$	the spherical idempotent and algebra	$\eK=\one_K/(q+1)$
$W_{\mathrm{aff}}$, $S_{\mathrm{aff}}$, $\Omega$, $H_{\mathrm{aff}}$	$\Waff$, $\Saff$, $\Om$, $\Haff$	same (§7.1)	$\Om=1$

Bernstein's presentation and the centre, restated Proved (cited)

(Bernstein presentation) [HKP10, Lemma 1.7.1, Rmk 1.7.2, (1.15.2)] Proved (cited)

(i)There is an injective algebra homomorphism $\C[\Xs]\to\cH$, $\lambda\mapsto\Th_\lambda$, such that for $\lambda=\lambda_1-\lambda_2$ with $\lambda_1,\lambda_2$ dominant, $$\Th_\lambda=q^{\langle\rho,\lambda_2-\lambda_1\rangle}\,T_{t^{\lambda_1}}\,T_{t^{\lambda_2}}^{-1},\qquad\text{in particular }\ \Th_\lambda=q^{-\langle\rho,\lambda\rangle}T_{t^{\lambda}}\ \text{ for dominant }\lambda,$$ where $\rho$ is half the sum of the positive roots.

(ii)Multiplication $\C[\Xs]\otimes\Hzero\to\cH$, $\lambda\otimes h\mapsto\Th_\lambda h$, is an isomorphism of vector spaces.

(iii)For a simple root $\alpha$ with reflection $s=s_\alpha$ and $\mu\in\Xs$: $$T_s\,\Th_\mu=\Th_{s(\mu)}\,T_s+(1-q)\,\frac{\Th_{s(\mu)}-\Th_\mu}{1-\Th_{-\alpha^\vee}} .$$ (The fraction is a polynomial in $\C[\Xs]$: $\mu-s(\mu)$ is a multiple of $\alpha^\vee$.)

The paper's proof, in outline

The module $M=\Cc(T(\cO)U(F)\backslash\GF/I)$ has basis $v_x$, $x\in\Wt$, and is free of rank one over $\cH$ with generator $v_1$ (Lemma 1.6.1: $h\mapsto v_1h$ is triangular with nonzero diagonal in the two bases, by a support computation using the Iwahori factorization). Hence $\cH=\End_{\cH}(M)$, and the commuting left action of $\C[\Xs]$ on $M$ embeds $\C[\Xs]$ in $\cH$; (1.7.1) $v_1\Th_\lambda=\pi^\lambda v_1$ characterizes $\Th_\lambda$, and (1.6.3) gives the formula in (i). (ii) is Lemma 1.7.1 by comparing bases. (iii) is derived from the rank-one computation of the intertwiner $I_{s_\alpha}$ on $M$ (Lemma 1.13.1, equation (1.14.1)), which reduces to $\SL_2$ (§1.12).

(the centre) [HKP10, Lemma 2.3.1] Proved (cited)

$$\Zc=\C[\Xs]^W .$$

The paper's proof, in outline

$\C[\Xs]^W$ is central by (iii) (Lemma 2.1.1). For the converse, extend scalars to the fraction field $L$ of $\C[\Xs]$: the normalized intertwiners $K_w$ make $L\otimes_{\C[\Xs]^W}\cH$ a matrix algebra over $L^W$ (Lemma 2.3.1), whose centre is $L^W$; intersecting with $\cH$ gives $\C[\Xs]^W$.

In the running example For $\SL_2$, $\alpha^\vee$ is dominant with $\langle\rho,\alpha^\vee\rangle=1$ and $t^{\alpha^\vee}=s_1s_0$ (Groups §4), so $\Th_{\alpha^\vee}=q^{-1}T_{s_1}T_{s_0}$ and $\Th_{-\alpha^\vee}=q\,(T_{s_1}T_{s_0})^{-1}=q\,T_{s_0}^{-1}T_{s_1}^{-1}$. The centre is spanned by $\Th_{n\alpha^\vee}+\Th_{-n\alpha^\vee}$, $n\ge0$. For $\PGL_2$ the extra element $\Th_{\omega^\vee}$ with $\Th_{\omega^\vee}^2=\Th_{\alpha^\vee}$ appears (SL₂ III).

The spherical Hecke algebra as a corner, and the Satake isomorphism Proved here

($\HK=\eK\cH\eK$) [HKP10, §4.1] Proved here

Let $\eK=\vol(K)^{-1}\one_K=[K:I]^{-1}\one_K\in\cH$. Then

$$\eK\ast\eK=\eK,\qquad \eK\ast f=f\ \text{ for left }K\text{-invariant }f,\qquad \HK=\Cc(K\backslash\GF/K)=\eK\cH\eK,$$

and the product of $\HK$ (for the Haar measure with $\vol(K)=1$) is the product of $\cH$ restricted to the corner.

Proof

$(\one_K\ast f)(x)=\sum_{y\in K/I}f(y^{-1}x)=[K:I]\,f(x)$ when $f$ is left $K$-invariant; with $f=\one_K$ this gives $\one_K\ast\one_K=[K:I]\one_K$, hence $\eK^2=\eK$ and $\eK\ast f=f$.
$\eK\ast g\ast\eK$ is $K$-invariant on both sides for any $g\in\cH$ (averaging over $K$ on each side), and conversely a $K$-bi-invariant $f$ equals $\eK\ast f\ast\eK$. So $\eK\cH\eK$ is the space of $K$-bi-invariant compactly supported functions.
For $f,g\in\HK$, the convolution with $\vol(I)=1$ is $[K:I]$ times the convolution with $\vol(K)=1$, since the measures differ by that factor; the paper's identification uses the measure with $\vol(K)=1$ on $\HK$ and the corner product $f\ast g$ computed in $\cH$ with $\vol(I)=1$ then rescaled. On this site $\HK$ is given the measure with $\vol(K)=1$, so that $\one_K$ is its unit, exactly as in the proof workspace [Draft].

(Satake isomorphism through $\cH$) [HKP10, §4.2–4.3, p. 11–12; (4.6.1)] Proved (cited)

The Satake transform $h\mapsto h^\vee$, characterized by $\one_{T(\cO)U(F)K}\ast h=h^\vee\cdot\one_{T(\cO)U(F)K}$ in $\Cc(T(\cO)U(F)\backslash\GF/K)$, is an algebra isomorphism $\HK\simeq\C[\Xs]^W=\Zc$, and for corresponding $h\in\HK$ and $z\in\Zc$,

$$h=\eK\,z .$$

The paper's proof, in outline

$M\eK=\Cc(T(\cO)U(F)\backslash\GF/K)$ is free of rank one over $\C[\Xs]$ on the spherical vector, which defines $h\mapsto h^\vee$; the normalized intertwiners fix the spherical vector, so $h^\vee$ is $W$-invariant; a triangularity argument on the basis $\one_{Kt^\mu K}$ shows bijectivity (§4.3). Then $m\eK z=mh$ for all $m\in M$ gives (4.6.1).

So the spherical algebra is a corner of $\cH$, not a subalgebra containing the unit: its unit is $\eK$, and $\eK\neq T_e$ as soon as $W\neq1$. The proof workspace's Part II turns on this distinction [Draft].

Dictionary to the paper's conventions, with its wording Context

Modules. The paper works with right $\cH$-modules ($M$ arises as $I$-fixed vectors of a right $G$-module) and passes to left modules by $g\mapsto g^{-1}$ (§1.8). This site and the proof workspace use left modules; the relations of the Concept are stated for left multiplication by $T_s$, the paper's display is for right multiplication, and the two are exchanged by the anti-involution $T_w\mapsto T_{w^{-1}}$.
Normalization of $\Th_\lambda$. The factor $q^{\langle\rho,\lambda_2-\lambda_1\rangle}$ depends on $\delta_B^{1/2}$; the paper notes that its $\Th_\lambda$ agrees with Lusztig's (Remark 1.7.2).
Measures. $\cH$ uses $\vol(I)=1$ and $\HK$ uses $\vol(K)=1$ (§4.1), as here.

Verbatim wording (arXiv v3 pages)

“1.2. Iwahori-Hecke algebra $H$. We denote by $H$ the Iwahori-Hecke algebra $C_c(I\backslash G/I)$. The convolution product is defined using the Haar measure giving $I$ measure 1. The elements $T_x:=1_{IxI}$ ($x\in\widetilde W$) form a $\mathbf C$-basis for $H$.”[HKP10, p. 2]

“The proof of (ii) above parallels the standard proof of the Iwahori-Matsumoto relations in $H$, which state that for $x\in\widetilde W$, $s\in S_{\mathrm{aff}}$, and $\sigma\in\Omega$: $T_xT_s=T_{xs}$ if $x<xs$; $T_xT_s=q\cdot T_{xs}+(q-1)\cdot T_x$ if $xs<x$; $T_xT_\sigma=T_{x\sigma}$, where $<$ denotes the Bruhat order determined by $S_{\mathrm{aff}}$.”[HKP10, §7.2, p. 20]

“4.1. Definition of $H_K$ and $M_K$. Let $e_K$ be the idempotent $1_K/\mathrm{meas}(K)$ in $H$. Put $H_K:=C_c(K\backslash G/K)$, which we identify with the subring $e_KHe_K$ of $H$ (so that $1_K\in H_K$).”[HKP10, p. 11]

“Our normalization is $\mu\mapsto\pi^\mu$, and therefore our Bruhat order on $X_*(A)\rtimes W$ is the one determined by the simple affine reflections about the walls of the unique alcove in $X_*(A)$ whose closure contains the origin and lies in the negative Weyl chamber.”[HKP10, footnote 2, p. 3]

What does not transfer to the present setting Context

The paper's field is $p$-adic; here $F=\Fq((t))$. Every argument used only needs a local field with finite residue field and a split group over its ring of integers, so nothing changes; but the statement is cited, not re-proved, for equal characteristic.
Macdonald's formula, the Casselman–Shalika formula and the Lusztig–Kato formula (§5–7) are not used on this site.
The paper's proof of the Iwahori–Matsumoto relations is only sketched (“the standard proof”, §7.2); the proof on the Concept is the one from Iwahori–Matsumoto with the counts written out.