$(\mathfrak{g},K)$-module of $\mathrm{O}(p,q)$ associated with the finite-dimensional representation of $\mathfrak{sl}_2$

The main aim of this paper is to construct irreducible $(\mathfrak{g},K)$-modules of $\mathrm{O}(p,q)$ corresponding to the finite-dimensional representation of $\mathfrak{sl}_2$ of dimension $m+1$ under the Howe duality, to find the $K$-type formula, the Gelfand-Kirillov dimension and the Bernstein degree of them, where $m$ is a non-negative integer. The $K$-type formula for $m=0$ shows that it is nothing but the $(\mathfrak{g},K)$-module of the minimal representation of $\mathrm{O}(p,q)$. One finds that the Gelfand-Kirillov dimension is equal to $p+q-3$ not only for $m=0$ but for any $m$ satisfying $m+3 \leq (p+q)/2$ when $p, q \geq 2$ and $p+q$ is even, and that the Bernstein degree for $m$ is equal to $(m+1)$ times that for $m=0$.

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