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Leonid Petrov
Leonid Petrov
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Let us take an N-dimensional (N odd) irreducible representation V of SL(2,R).

It is known that (e.g., Lie groups and Lie algebras III by Vinberg and Onischik, 1994 p. 94) in V there is an invariant symmetric bilinear form $b$ for the action of SL(2,R). Thus, SL(2,R) is embedded into $O(V,b)$ - the orthogonal group of V with respect to the form $b$.

Consider a spin representation of $O(V,b)$, this is a representation in the space of dimension $2^N$. One can restrict this representation to $SL(2,R)\subset O(V,b)$.

The question is: how to decompose this representation of SL(2,R) of dimension $2^N$ into irreducibles?

Let us take an N-dimensional (N odd) irreducible representation V of SL(2,R).

It is known that (e.g., Lie groups and Lie algebras III by Vinberg, 1994 p. 94) in V there is an invariant symmetric bilinear form $b$ for the action of SL(2,R). Thus, SL(2,R) is embedded into $O(V,b)$ - the orthogonal group of V with respect to the form $b$.

Consider a spin representation of $O(V,b)$, this is a representation in the space of dimension $2^N$. One can restrict this representation to $SL(2,R)\subset O(V,b)$.

The question is: how to decompose this representation of SL(2,R) of dimension $2^N$ into irreducibles?

Let us take an N-dimensional (N odd) irreducible representation V of SL(2,R).

It is known that (e.g., Lie groups and Lie algebras III by Vinberg and Onischik, 1994 p. 94) in V there is an invariant symmetric bilinear form $b$ for the action of SL(2,R). Thus, SL(2,R) is embedded into $O(V,b)$ - the orthogonal group of V with respect to the form $b$.

Consider a spin representation of $O(V,b)$, this is a representation in the space of dimension $2^N$. One can restrict this representation to $SL(2,R)\subset O(V,b)$.

The question is: how to decompose this representation of SL(2,R) of dimension $2^N$ into irreducibles?

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Leonid Petrov
Leonid Petrov
  • 1.8k
  • 12
  • 25

A decomposition of the "spin representation" of SL(2)

Let us take an N-dimensional (N odd) irreducible representation V of SL(2,R).

It is known that (e.g., Lie groups and Lie algebras III by Vinberg, 1994 p. 94) in V there is an invariant symmetric bilinear form $b$ for the action of SL(2,R). Thus, SL(2,R) is embedded into $O(V,b)$ - the orthogonal group of V with respect to the form $b$.

Consider a spin representation of $O(V,b)$, this is a representation in the space of dimension $2^N$. One can restrict this representation to $SL(2,R)\subset O(V,b)$.

The question is: how to decompose this representation of SL(2,R) of dimension $2^N$ into irreducibles?