| ZHONG JIAQING.[J].数学年刊A辑,1980,1(2):261~272 |
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| DIMENSIONS OF THE RING OF INVARIANT OPERATORSON BOUNDED HOMOGENEOUS DOMAINS |
| Received:December 13, 1978 |
| DOI： |
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| 英文摘要: |
| Let \[{\cal D}\] be the bounded homogeneous domain, \[G({\cal D})\]be the group of the auto-
morphism of \[{\cal D}\]. A differential operator T of (k,l) type is called invariant, if and
only if it is commutative with the actions of the \[G({\cal D})\] . The ring of the (k,l) type
invariant differential operators is denoted by \[{I_{k,l}}({\cal D})\].
This paper gives a formula for caculating the dimension of \[{I_{k,l}}({\cal D})\]. Obviously,
these dimensions are the invarians of \[{\cal D}\] in analytical equivalence,
By means 〇f the results in the famous work of A,, Selberg4, and applying the
theory of group representations, we have proved the following
Theorem.
\[\dim {I_{k,l}}({\cal D})/\sum\limits_{({k^'},{l^'}) < (k,l)} {{I_{{k^'},{l^'}}}({\cal D})} = \int_K^{} {\cal X} (k,0,...,0)(A)\overline {{\cal X}(k,0,...,0)(A)} \dot A\]K
where K is the group formed by the linear parts of the isotropy of \[G({\cal D})\] at \[0 \in {\cal D}\]
and \[{{\cal X}(k,0,...,0)(A)}\] is the character of the irreducible representation of the unitary group
U(n) with Young index (k, 0, ???, 0).
Using this formula, we caculate the dimensions for some concrete cases, including
those with the classical symmetric domain, and with a Siegel domain that is not
symmetric. |
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