Disjointness of representations arising in harmonic analysis on the infinite-dimensional unitary group.

*(English. Russian original)*Zbl 1213.43011
Funct. Anal. Appl. 44, No. 2, 92-105 (2010); translation from Funkts. Anal. Prilozh. 44, No. 2, 14-32 (2010).

The article is devoted to representations of the infinite-dimensional unitary group \(U(\infty)\), that is the union of classical groups \(U\left(N\right)\) naturally embedded in one another. Since this group is not locally compact, the conventional definition of the regular representation cannot be applied to it. So, the author deals with a natural generalization of the regular representation of the group \(U\left(\infty\right)\) introduced by G. Olshanski [J. Funct. Anal. 205, No. 2, 464–524 (2003; Zbl 1036.43002)].

The space of representations introduced by G. Olshanski is an inductive limit of the spaces \(L_2(U(N))\). For this limit to be well-defined, one should choose embeddings \(L_2(U(N))\hookrightarrow L_2(U(N+1))\). Since this choice is not unique, different possible choices lead to a family of representations \(T_{z,w}\) which depend on two complex parameters, \(z\) and \(w\). The representation \(T_{z,w}\) will not change if \(z\) or \(w\) is replaced by \(\overline{z}\) or \(\overline{w}\), respectively. The structure of the representation heavily depends on whether the parameters are integer or not. The article deals with the latter case.

The very interesting question is: how different are the representations \(T_{z,w}\) despite the fact that they all are inductive limits of the same sequence of spaces? The author answers this question by proving that the representations \(T_{z,w}\) are pairwise disjoint, i.e., have no equivalent nonzero subrepresentations. The main result of the article can be expressed by the following theorem.

Theorem. Suppose that the parameters \(z\), \(w\), \(z'\) and \(w'\) are non-integer, \(\{z,\overline{z}\}\neq\{z', \overline{z'}\}\), \(\{w,\overline{w}\}\neq\{w',\overline{w'}\}\) (all pairs are unordered), and \(\text{Re}(z+w)>-1/2\). Then the representations \(T_{z,w}\) and \(T_{z',w'}\) are disjoint.

As the author notes, decompositions of the representations \(T_{z,w}\) into irreducible representations are determined by certain spectral measures. To prove the disjointness of the representations, it is sufficient to prove this for the corresponding spectral measures. The author reduces the disjointness of the spectral measures to that of certain probability measures \(\rho_{z,w}\) on the space of paths in the Gelfand-Tsetlin graph. These measures are the main object of study in the article.

The article also contains some interesting applications of the methods developed. A path in the Gelfand-Tsetlin graph is a sequence \(\{\lambda(N)\}\) of dominant weights of irreducible representations of the unitary groups \(U(N)\), and each \(\lambda(N)\) can be regarded as the pair \((\lambda^+(N),\lambda^-(N))\) of Young diagrams. Thus, each measure \(\rho_{z,w}\) generates two random sequences, \(\{\lambda^+(N)\}\) and \(\{\lambda^-(N)\}\), of Young diagrams, which form a Markov growth process. Besides, the author also proves that the length of the diagonal in \(\{\lambda^\pm(N)\}\) has at most logarithmical growth in \(N\).

The article also contains an interesting introduction, comparison with earlier results, an extended bibliography, and some unsolved problems that could encourage mathematicians to new investigations in the area. It should be very interesting for specialists in abstract harmonic analysis.

The space of representations introduced by G. Olshanski is an inductive limit of the spaces \(L_2(U(N))\). For this limit to be well-defined, one should choose embeddings \(L_2(U(N))\hookrightarrow L_2(U(N+1))\). Since this choice is not unique, different possible choices lead to a family of representations \(T_{z,w}\) which depend on two complex parameters, \(z\) and \(w\). The representation \(T_{z,w}\) will not change if \(z\) or \(w\) is replaced by \(\overline{z}\) or \(\overline{w}\), respectively. The structure of the representation heavily depends on whether the parameters are integer or not. The article deals with the latter case.

The very interesting question is: how different are the representations \(T_{z,w}\) despite the fact that they all are inductive limits of the same sequence of spaces? The author answers this question by proving that the representations \(T_{z,w}\) are pairwise disjoint, i.e., have no equivalent nonzero subrepresentations. The main result of the article can be expressed by the following theorem.

Theorem. Suppose that the parameters \(z\), \(w\), \(z'\) and \(w'\) are non-integer, \(\{z,\overline{z}\}\neq\{z', \overline{z'}\}\), \(\{w,\overline{w}\}\neq\{w',\overline{w'}\}\) (all pairs are unordered), and \(\text{Re}(z+w)>-1/2\). Then the representations \(T_{z,w}\) and \(T_{z',w'}\) are disjoint.

As the author notes, decompositions of the representations \(T_{z,w}\) into irreducible representations are determined by certain spectral measures. To prove the disjointness of the representations, it is sufficient to prove this for the corresponding spectral measures. The author reduces the disjointness of the spectral measures to that of certain probability measures \(\rho_{z,w}\) on the space of paths in the Gelfand-Tsetlin graph. These measures are the main object of study in the article.

The article also contains some interesting applications of the methods developed. A path in the Gelfand-Tsetlin graph is a sequence \(\{\lambda(N)\}\) of dominant weights of irreducible representations of the unitary groups \(U(N)\), and each \(\lambda(N)\) can be regarded as the pair \((\lambda^+(N),\lambda^-(N))\) of Young diagrams. Thus, each measure \(\rho_{z,w}\) generates two random sequences, \(\{\lambda^+(N)\}\) and \(\{\lambda^-(N)\}\), of Young diagrams, which form a Markov growth process. Besides, the author also proves that the length of the diagonal in \(\{\lambda^\pm(N)\}\) has at most logarithmical growth in \(N\).

The article also contains an interesting introduction, comparison with earlier results, an extended bibliography, and some unsolved problems that could encourage mathematicians to new investigations in the area. It should be very interesting for specialists in abstract harmonic analysis.

Reviewer: Alexander Tovstolis (Donetsk)

##### MSC:

43A65 | Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) |

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\textit{V. E. Gorin}, Funct. Anal. Appl. 44, No. 2, 92--105 (2010; Zbl 1213.43011); translation from Funkts. Anal. Prilozh. 44, No. 2, 14--32 (2010)

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