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Функции Уиттекера с одномерным k-типом на полупростой группе ли эрмитова типа.

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Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
MSC 43A85
Whittaker functions with one-dimensional
K -type on a semisimple Lie group
of Hermitian type
c
N. Shimeno
Okayama University of Science, Okayama, Japan
We give an explicit formula for the Whittaker function with one-dimensional K -type on a
simple Lie group of Hermitian type.
Keywords:
semisimple Lie groups, Hermitian symmetric spaces, K -types, Harish-Chandra
c-function, Whittaker functions
Introduction
The radial part of the zonal spherical function ?? (a) on a Riemannian symmetric
space can be written in the form
X
c(w?)?w? (a)
(1)
?? (a) =
w?W
for generic ?, where W is the Weyl group, c(?) is the Harish-Chandra c-function,
and ?? (a) is the joint eigenfunction of the radial parts of the invariant dierential
operators given by a series expansion ?? (a) ? a??? as a ? ?. The coecients of
the series expansion are determined recursively from the equation for the radial part
of the Casimir operator.
Hashizume [2] proved a formula that is similar to (1) for the class one Whittaker
function given by the Jacquet integral on a real semisimple Lie group. In this
article, we generalize the result of Hashizume for the Whittaker function with onedimensional K -type on a simple Lie group of Hermitian type. We also give the
formula for the radial part of the Casimir operator for the Siegel-Whittaker function
on a Hermitian symmetric space of tube type.
We will give the proof in a forthcoming paper.
1759
Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
џ 1. JacquetWhittaker functions
Let G/K be an irreducible Hermitian symmetric space with G a simple Lie group
of Hermitian type and K a maximal compact subgroup of G. Let G = N AK be
a Iwasawa decompostion. Let g, n, a, k denote the Lie algebras of G, N , A, K
respectively. Fix an inner product h , i on a.
Let ? denote the restricted root system for G/K , and ?+ the positive system
+
corresponding to N . Let ? ? ?+ denote the
P set of simple roots in ? . Let m? be
the multiplicity of ? ? ? and ? = (1/2) ???+ m? ?. Let W be the Weyl group
for ? and s? ? W denote the simple reection corresponding to ? ? ?. Let `(w)
denote the length of w ? W , w0 ? W the longest element of W , and w0 ? NK (a) a
representative of w0 .
Let ? be a normalized nondegenerate unitary character of N and ?` a unitary
character of K . Here ` ? Z if G is a real form of the simply connected complex Lie
group with the Lie algebra gC , and ` ? R if G is simply connected. Let ? ? a?C .
Dene the function 1?,` on G = N AK by
1?,` (nak) = a?+? ??` (k),
n ? N, a ? A, k ? K.
Dene the Jacquet-Whittaker function with one-dimensional K -type by
Z
W` (?, ?; g) =
?(n)?1 1?,` (w0?1 ng) dn,
N
where dn is a suitably normalized invariant measure on N (cf. [4], [7], [2], [1]).
The following properties characterize the meromorphic continuation in ? of
u(a) = W` (?, ?; a):
(i) u(nak) = ?(n) u(a) ??` (k), n ? N, a ? A, k ? K ;
(ii) D u = ?` (D)(?)u, ? D ? D` (G/K), where D` (G/K) is the algebra of
invariant dierential operators on the homogeneous line bundle over G/K associated
with ?` and ?` : D` (G/K) ? S(a)W the Harish-Chandra isomorphism;
(iii) u(a) is of moderate growth, i.e. |u(a)| 6 C exp{k| log a|}, C > 0, k > 0;
(iv) u(a) ? ?` (w0 ) c` (?) aw0 ?+? as a ? ?.
For the root system of type BCr , there are three root length, say long, middle,
and short. If G/K is of tube type, then the root system is of type Cr and there are
no short roots. The Harish-Chandra c-function for the one-dimensional K -type ?`
is given by
2??? ?(?? )
Ч
1
1
1
1
+
??? , middle ?
(?? + m? + 1) ?
(?? + m? )
2
2
2
2
??
Y
2 ?/2 ?(??/2 )
,
Ч
1
1
1
1
???+ , long ?
(??/2 + m?/2 + 1 + `) ?
(??/2 + m?/2 + 1 ? `)
2
2
2
2
c` (?) = c1
1760
Y
Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
h?, ?i
and c1 is determined by c0 (?) = 1 (cf. [8], [9]).
h?, ?i
The set of the simple roots is given by
where ?? =
? = {er ? er?1 , . . . , e2 ? e1 , 2e1 } (tube type),
? = {er ? er?1 , . . . , e2 ? e1 , e1 } (non-tube type),
where r is the rank of the symmetric space. There are three root length, long (2ei ),
middle (ei ± ej ), and short (ei ).
Dene numbers M` (?, ? ; w), w ? W , recursively by
M` (?, ? ; e) = 1,
M` (?, ? ; w s? ) = M` (?, ? ; w) M` (w?, ? ; s? ),
`(w s? ) = `(w) + 1.
Then M` (?, ? ; s? ), ? ? ?, are given according to the length of ? as follows (cf. [4],
[7], [1], [2]):
(i) If ? ? ? is a middle root, then
M` (?, ? ; s? ) = 2?4?? e? (?) e? (??)?1 ,
1
1
?1
(?? + m? + 1) ?
(?? + m? ) .
e? (?) = ?
2
2
(ii) If ? = e1 ? ? is a short root (the case of non-tube type), then
M` (?, ? ; s? ) = 2?4?? e? (?, `) e? (??, `)?1 ,
1
1
1
1
?1
e? (?, `) = ?
(?? + m? + 1 + `) ?
(?? + m? + 1 ? `) .
2
2
2
2
(iii) If ? = 2e1 ? ? is a long root (the case of tube type), then
1 `
? ??? + ?
2 2
?2??
.
M` (?, ? ; s? ) = 2
1 `
? ?? + ?
2 2
Then the Whittaker function W` (?, ? ; g) satises the functional equation
W` (?, ? ; g) = M` (?, ? ; w) W` (w?, ? ; g),
w ? W.
Let ?(?) denote the radial part of the Casimir operator ? with respect to the
Iwasawa decomposition G = N AK for representations ? and ?` from the left and
right respectively. Then e?? ? (?(?) + h?, ?i) ? e? is given by
X
?a ?
h?, ?i e2? ? 4he1 , e1 i ` e2e1 (tube type),
???
?a ?
X
h?, ?i e2?
(non-tube type),
???
1761
Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
where ?a is the Casimir operator on a. The operator is the same as the class one
case for non-tube type, and there is an additional term for tube type:
e
??
r
r?1
X
?2
1 X ti+1 ?ti
? (?(?) + h?, ?i) ? e =
?
e
? e2t1 ? ` et1 .
2
?t
2
i
i=1
i=1
?
(2)
Radial parts of invariant dierential operators give a family of commuting
dierential operators, which prove complete integrability of Toda model with the
above Schrodinger operator.
Let ?? (a) denote the series solution of
(e?? ? (?(?) + h?, ?i) ? e? )?? = h?, ?i??
of the form
?? (a) = a?
X
bµ (?)aµ ,
b0 (?) = 1,
µ??
P
where ? denotes the family of linear combinations
n? ? with ? ? ? and n? ? Z+ .
?
Then e ?? is an eigenfunction of ?(D) for all D ? Dl (G/K), and functions ?w? ,
w ? W , form a basis of the space of the joint eigenfunctions for generic ?.
Now we state the main result of this article.
Theorem 1
?
For generic
W` (?, ? ; a) =
we have
X
M` (?, ? ; w0 w) c` (w0 w?) ?(w?, ? ; a).
w?W
If ` = 0, the above theorem is a special case of the result of Hashizume [2].
As in the case of the Whittaker functions on semisimple Lie groups of rank
one, Whittaker functions depend essentially on reduced root system consisting of
inmultiplicative roots in ? and ? (and ` for tube type). We dene the Whittaker
function associated with Cr -type root system. For G = Sp(r, R), the system ? is of
type Cr and m? = 1 for all ? ? ?. Dene a function W?,` (a) on A by
W?,` (a) = d` (?)?1 a?? W` (?, ? ; a)
where
d` (?) = ?` (w0 ) c` (?)
Y
2??? ?(?? )?1 Ч
???+ , middle
Y
Ч
???+ , long
??
2
1 `
? ?? + +
?(2?? )?1
2 2
(3)
with c` (?) the c-function for Sp(n, R)/U(r). Then W?,` (a) is W -invariant with
respect to ? and
lim a?w0 ? W?,` (a) = d` (?)?1 c` (?).
a??
1762
Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
Corollary 1
Let
G/K
be an irreducible Hermitian symmetric space of tube type.
Then we have
W` (?, ? ; a) = d` (?)a? W(?, ` ; a),
where
d` (?)
is given by
(3)
with
c` (?)
the
c-function
for
G/K .
We can regard W?,` (a) as a multivariable analogue of the classical Whittaker
function W?,µ (z), which is associated to the root system of type Cr . In a similar
way, we can dene a multivariable analogues of the modied Bessel function of the
second kind associated with reduced root systems from class one Whittaker functions
on real split simple Lie groups.
џ 2. Siegel-Whittaker functions
Let G/K be an irreducible Hermitian symmetric space of tube type, Ps = Ls nNs
Siegel parabolic subgroup of G, R = (Ls ? K) n Ns . Then we have the generalized
Cartan decomposition G = RAK . Let ? be a normalized unitary character of Ns
that is xed by Ls ? K . We consider functions on G that satisfy
u(rak) = trivLs ?K · ?(r)?1 u(a) ??` (k),
r ? R, a ? A, k ? K.
Then the radial part ?(?) of the Casimir operator ? for such u satises
? 1/2 ? (?(?) + h?, ?i) ? ? ?1/2
r
r
X mej ?ei (2 ? mej ?ei )
X
X
?2
+
?
4
(e4ti + `e2ti ),
=
2
2
?t
2 sinh (tj ? ti )
i
16i<j6r
i=1
i=1
where
?=
Y
???+ ,middle
m?
(sinh ?)
r
Y
(4)
e?ti /2 .
i=1
Radial parts of invariant dierential operators give a commuting family of dierential
operators containing the above Schr
odinger operator, which prove integrability of
the model.
There exists a unique joint eigenfunction globally dened on A and of moderate
growth up to constant multiples ([11]). This function was studied by Ishii [3] for
G = SO0 (2, n).
Remark 1
(i) It is known that the Schrodinger operators (2) and (4) give
quantum integrable models (cf. [5]). Radial parts of invariant dierential operators
give a family of commuting dierential operator, which prove complete integrability
of the models with the Schrodinger operators (2) and (4). These group theoretic
interpretations seem to be new.
1763
Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
(ii) We can prove in a similar way as [10] that the C -type Whittaker function
W?,` (a) is a degenerate limit of the BC -type Heckman-Opdam hypergeometric
function. The Siegel-Whittaker functions of moderate growth is also a degenerate
limits of the Heckman-Opdam hypergeometric functions (cf. [6]).
References
1. R. Goodman and N. R. Wallach. Whittaker vectors and conical vectors. J.
Funct. Anal., 1980, vol. 39, 199279.
2. M. Hashizume. Whittaker functions on semisimple Lie groups, Hiroshima
Math. J., 1982, vol. 12, 259293.
3. T. Ishii. Siegel-Whittaker functions on SO0 (2, q) for class one principal series
representations, Compos. Math., 2004, vol. 140, 827854.
4. H. Jacquet. Fonctions de Whittaker associees aux groupes de Chevalley, Bull.
Soc. Math. France, 1967, tome 95, 243309.
5. T. Oshima. Completely integrable quantum systems associated with classical
root systems, SIGMA, 3-071, 2007.
6. T. Oshima and N. Shimeno. HeckmanOpdam hypergeometric functions and
their specializations, preprint.
7. G. Schimann. Integrale d'entrelacement et fonctions de Whittaker, Bull. Soc.
Math. France, 1971, tome 99, 372.
8. H. Schlichtkrull. One-dimensional K-types in nite dimensional
representations of semisimple Lie groups: A generalization of Helgason's theorem,
Math. Scand., 1984, vol. 54, 279294.
9. N. Shimeno. Eigenspaces of invariant dierential operators on a homogeneous
line bundle on a Riemannian symmetric space, J. Fac. Sci. Univ. Tokyo, Sect. IA,
Math., 1990, vol. 37, 201234.
10. N. Shimeno. A limit transition from the HeckmanOpdam hypergeometric
functions to the Whittaker functions associated with root systems, arxiv:0812.3773.
11. H. Yamashita. Finite multiplicity theorems for induced representations
of semisimple Lie groups. II. Applications to generalized GelfandGraev
representations, J. Math. Kyoto Univ., 1988, vol. 28, 383444.
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