Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
Last update: 27 September 2012
The Satake isomorphism
The following theorem establishes a -analogue of the isomorphism from Proposition 2.3(c). The map
in the following theorem is the Satake isomorphism. We shall continue to abuse language and use the
term "vector space" in place of "free module".
The vector space isomorphism in Proposition 2.3(c) generalizes to a vector space isomorphism
Proof.
Using the third equality in (2.5),
By Proposition 2.3(c) and Theorem 1.4,
and so
Since
is a basis of
and
is a basis of
the composite map
is a vector space isomorphism.
If let
In particular, if then
and
is the polynomial that appears in (2.1).
The Hall-Littlewood polynomials, or Macdonald spherical functions, are defined by
Then
and, using the Weyl denominator formula,
and so, conceptually, the spherical functions
interpolate between the Schur functions and the monomial symmetric functions
The double cosets in
are
If
let and
be the maximal and minimal length elements of
respectively. Theorem 2.9 below will show that under the Satake isomorphism the Weyl characters correspond to
Kazhdan Lusztig basis elements and the polynomials
correspond to the elements
More precisely, we have the following diagram:
where is the longest element of The following three lemmas
(of independent interest) are used in the proof of Theorem 2.9.
Let
be commuting variables indexed by the positive roots. For
let
be as in (2.9),
as in (2.8), and define
and
where, as in (1.5),
is the inversion set of Then
with
unless
and
with
unless
and
Proof.
(a) If let
and let be as defined in (2.4). Using the Weyl denominator formula, Proposition 2.3(a), and the second equality in (2.5),
which shows that is a symmetric function and
By the straightening law for Weyl characters (2.7),
or
with
Let denote the complement of in
Since permutes the elements of
for some subset (which could be determined explicitly in terms of and Hence
This proves that unless
In (2.12), only if
and
in which case
Thus
(b) Set for all
Applying (a) with
Let be a set of minimal length coset representatives of the cosets in
Every element
can be written uniquely as with
and
(see [Bou1981, IV §1 Ex. 3]). Let
and let be the complement of
in
Then permutes the elements of
among themselves and so
where the last equality follows from (2.13). Thus there is an element
where is the field of fractions of such that
Since is a symmetric polynomial (an element of
Since only occurs in the numerators of the terms in the sum defining in fact
is a symmetric polynomial with coefficients in
It follows that all the
appearing in part (a) are divisible by
and
are polynomials in such that
and
unless
Lemma 2.5 has the following interesting (and useful) corollary, see [Mac1972].
Let and be as in (1.8) and (1.1), respectively, and let
be as defined in (2.8).
Proof.
Part (a) follows from Lemma 2.5 (a) by setting and specializing
for all
(b) Applying the homomorphism
to both sides of the identity in (a) for the root system
gives
If and
is a reduced word for then
and so
Then
Thus the only nonzero term on the right hand side of (2.14) occurs for
For let
be the translation in
and let be the maximal length element in the double coset
Let
as in (2.4). Then
in the affine Hecke algebra
Proof.
Let Let
and let and be the maximal length elements in
and respectively. Let
and be the minimal and maximal length elements respectively in the double coset
For each positive root the
hyperplanes
are between the fundamental alcove and the alcove and so
Since
and
For example, in the setting of Example 1.1, if
in type then
and
Labelling the alcove by the element the 32 alcoves
with
make up the four shaded diamonds
Then
Let be a set of minimal length coset representatives of the cosets in
Every element
has a unique expression with
and
If
then
Since conjugation by and conjugation by are automorphisms
of and respectively taking simple reflections to simple reflections,
Then
Let be the longest element of and let
Proof.
(a) If then
and
Thus,
Let and write
with
Since
and
(b) For
These are the defining properties (before 2.1) of and
and so and
(c) If
then, since
and, by (a),
since is
-invariant. Finally, since
is central, and
(d) By (a), (b) and the third equality in (2.5),
The following theorem is due to Lusztig [Lus1983]. Part (a) was originally proved in a different formulation by Macdonald [Mac1971, (4.1.2)].
If let be the stabilizer of and let
be as in (2.8).
Let Let
be the Macdonald spherical function
defined in (2.9) and define
as in (2.4). In the affine Hecke algebra
For let
be the translation in
and let be the maximal length element in the double coset
Let
be the Weyl character and let be the Kazhdan-Lusztig basis
element as defined in (2.6) and (1.26), respectively. In the affine Hecke algebra
Proof.
(a) By Theorem 2.4 there is an element
such that
To find first do a rank 1 calculation,
Since is a linear combination of products of it can also be written as a linear combination
of products of Thus
can be
written as a linear combination of terms of the form
Thus
and the are some linear combinations of products of terms of the form
for roots Since
is an element of
where is the longest element of The coefficient
comes from the highest term in the expansion of
in terms of linear combination of products of the
If
is a reduced word for then
by Lemma 1.2 and the fact that
Thus, since
(b) Since
Lemma 2.8 gives
By Lemma 2.5(b),
where
unless and
Thus, by part (a) and Lemma 2.7
where the polynomials
are 0 unless and
Hence
is a bar invariant element of such that its expansion in terms of the basis
is triangular with coefficient of equal to 1 and all other coefficients in
These are the defining properties (1.26)-(1.27) of
Acknowledgements
The research of A. Ram was partially supported by the National Science Foundation (DMS-0097977), the National Security Agency (MDA904-01-1-0032) and by EPSRC
Grant GR K99015 at the Newton Institute for Mathematical Sciences. The research of K. Nelsen was partially supported by the National Science Foundation (DMS-0097977
and a VIGRE grant) and the National Security Agency (MDA904-01-1-0032).