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**An analogue of the Borel-Weil-Bott theorem for Hermitian symmetric pairs of non-compact type.**
*(English)*
Zbl 0257.22013

The Theorem of Borel-Weil-Bott states that all irreducible representations of a compact connected real Lie group can be obtained as representations acting on sheaf cohomology groups associated to \(G\) and it is a conjecture of Langlands that all representations belonging to the discrete series of a noncompact connected semisimple real Lie group might be realized on “\(L^2\)-cohomology” groups. Under the conditions that \(G\) admits a finite dimensional faithful representation and that \(G/K\) is Hermitian symmetric, \(K\) being a maximal compact subgroup, the authors prove that for most discrete classes the conjecture holds.

The contents of the paper are clearly described in the introduction. “Let \(\mathfrak g\) and \(\mathfrak k\) be the Lie algebras of \(G\) and \(K\), respectively. Let \(\mathfrak h\) be a Cartan subalgebra of \(\mathfrak k\). Then \(\mathfrak h\) is also a Cartan subalgebra of \(\mathfrak g\). Choose an ordering on the roots of \((\mathfrak g^c, \mathfrak h^c)\) compatible with the complex structure on \(G/K\). For an irreducible unitary representation \(\tau_\Lambda\) of \(K\) with the highest weight \(\Lambda\), we denote by \(E_\Lambda\) the holomorphic vector bundle on \(G/K\) associated to the contragredient representation. Let \(H^{0,q}_2 (E_\Lambda)\) denote the Hilbert space of square-integrable harmonic forms of type \((0,q)\) with coefficients in \(E_\Lambda\) (these are the “square-integrable \(\bar\partial\)-cohomology spaces attached to \(E_\Lambda\) defined in [K. Okamoto and H. Ozeki, Osaka J. Math. 4, 95–110 (1967; Zbl 0173.23702)]. The unitary representation \(\pi^q_\Lambda\) of \(G\) on \(H^{0,q}_2 (E_\Lambda)\) decomposes into a finite number of irreducible representations each of which belongs to the discrete series for \(G\). Let \(\rho\) denote the half sum of positive roots of \((\mathfrak g^C, \mathfrak h^C)\) and let \(q_\Lambda\) denote the number of non-compact positive roots \(\alpha\) such that \(\langle \Lambda + \rho, \alpha\rangle >0\). We prove that the alternating sum of the characters of \(\pi^q_\Lambda\) is equal to \((-1)^{q_\Lambda}\Theta_{\omega(\Lambda + \rho)^*}\) where \(\omega(\Lambda + \rho)^*\) denotes the class of the discrete series contragredient to the one associated with the form \(\Lambda + \rho\) and \(\Theta_{\omega(\Lambda + \rho)^*}\) its character. This shows that, in any case, all the discrete classes are realized on a subspace of \(H^{0,q}_2 (E_\Lambda)\) for some \(\Lambda\) and some \(q\). We define a constant \(c\) and we prove that \(H^{0,q}_2 (E_\Lambda) =0\) if \(q\ne q_\Lambda\) provided that \(\vert \langle \Lambda + \rho, \alpha\rangle \vert > c\) for any non-compact positive root \(\alpha\). The alternating sum formula, together with the vanishing theorem, shows that, if \(\Lambda\) satisfies the above condition, the class of the representation \(\pi^{q_\lambda}_\Lambda\) on \(H^{0,q_\lambda}_2 (E_\Lambda)\) is precisely \(\omega(\Lambda + \rho)^*\).

We now briefly outline the proof of the alternating sum formula. Let \(L^{0,q}_2 (E_\Lambda)\) be the space of square-integrable forms of type \((0,q)\) with coefficients in \(E_\Lambda\). The group \(G\) acts, via the induced representation, on this space and let \(L^{0,q}_2 (E_\Lambda)_d\) be the discrete part of this space. The representation of \(G\) on \(L^{0,q}_2 (E_\Lambda)_d\) decomposes into a finite sum of discrete classes. The Laplacian \(\square\) has only finitely many eigenvalues on \(L^{0,q}_2 (E_\Lambda)\) and the sum of the eigenspaces coincides with \(L^{0,q}_2 (E_\Lambda)_d\). We then have a complex \[ \cdots \longrightarrow L^{0,q}_2 (E_\Lambda)_d \xrightarrow{\bar\partial} L^{0,q+1}_2 (E_\Lambda)_d \longrightarrow \cdots \] and the cohomology of this complex at the \(q\)-th stage is isomorphic to \(H^{0,q}_2 (E_\Lambda)\). If \(\varphi\) is a \(K\)-finite \(C^\infty\)-function with compact support on \(G\), \(\varphi\) defines an operator \(T_\varphi^q\) on \(L^{0,q}_2 (E_\Lambda)_d\) and \(\{T_\varphi^q\}\) forms an endomorphism of finite rank of the above complex. So it suffices to calculate the alternating sum of the traces of the operators \(T_\varphi^q\) on \(L^{0,q}_2 (E_\Lambda)_d\). The alternating sum of the traces of \(T_\varphi^q\) is given by an integral over the union of conjugacy classes of a compact Cartan subgroup. The integral involves the character of the representation of \(K\) and the integral is evaluated by using Weyl’s character formula for irreducible representations of \(K\) and coincides with Harish-Chandra’s character formula for the discrete series for \(G\).”

This paper stimulated further research, see R. Hotta [J. Math. Soc. Japan 23, 384–407 (1971; Zbl 0213.13701)], W. Schmid [Ann. Math. (2) 93, 1–42 (1971; Zbl 0291.43013)], R. Parthasarathy [Ann. Math. (2) 96, 1–30 (1972; Zbl 0249.22003)].

The contents of the paper are clearly described in the introduction. “Let \(\mathfrak g\) and \(\mathfrak k\) be the Lie algebras of \(G\) and \(K\), respectively. Let \(\mathfrak h\) be a Cartan subalgebra of \(\mathfrak k\). Then \(\mathfrak h\) is also a Cartan subalgebra of \(\mathfrak g\). Choose an ordering on the roots of \((\mathfrak g^c, \mathfrak h^c)\) compatible with the complex structure on \(G/K\). For an irreducible unitary representation \(\tau_\Lambda\) of \(K\) with the highest weight \(\Lambda\), we denote by \(E_\Lambda\) the holomorphic vector bundle on \(G/K\) associated to the contragredient representation. Let \(H^{0,q}_2 (E_\Lambda)\) denote the Hilbert space of square-integrable harmonic forms of type \((0,q)\) with coefficients in \(E_\Lambda\) (these are the “square-integrable \(\bar\partial\)-cohomology spaces attached to \(E_\Lambda\) defined in [K. Okamoto and H. Ozeki, Osaka J. Math. 4, 95–110 (1967; Zbl 0173.23702)]. The unitary representation \(\pi^q_\Lambda\) of \(G\) on \(H^{0,q}_2 (E_\Lambda)\) decomposes into a finite number of irreducible representations each of which belongs to the discrete series for \(G\). Let \(\rho\) denote the half sum of positive roots of \((\mathfrak g^C, \mathfrak h^C)\) and let \(q_\Lambda\) denote the number of non-compact positive roots \(\alpha\) such that \(\langle \Lambda + \rho, \alpha\rangle >0\). We prove that the alternating sum of the characters of \(\pi^q_\Lambda\) is equal to \((-1)^{q_\Lambda}\Theta_{\omega(\Lambda + \rho)^*}\) where \(\omega(\Lambda + \rho)^*\) denotes the class of the discrete series contragredient to the one associated with the form \(\Lambda + \rho\) and \(\Theta_{\omega(\Lambda + \rho)^*}\) its character. This shows that, in any case, all the discrete classes are realized on a subspace of \(H^{0,q}_2 (E_\Lambda)\) for some \(\Lambda\) and some \(q\). We define a constant \(c\) and we prove that \(H^{0,q}_2 (E_\Lambda) =0\) if \(q\ne q_\Lambda\) provided that \(\vert \langle \Lambda + \rho, \alpha\rangle \vert > c\) for any non-compact positive root \(\alpha\). The alternating sum formula, together with the vanishing theorem, shows that, if \(\Lambda\) satisfies the above condition, the class of the representation \(\pi^{q_\lambda}_\Lambda\) on \(H^{0,q_\lambda}_2 (E_\Lambda)\) is precisely \(\omega(\Lambda + \rho)^*\).

We now briefly outline the proof of the alternating sum formula. Let \(L^{0,q}_2 (E_\Lambda)\) be the space of square-integrable forms of type \((0,q)\) with coefficients in \(E_\Lambda\). The group \(G\) acts, via the induced representation, on this space and let \(L^{0,q}_2 (E_\Lambda)_d\) be the discrete part of this space. The representation of \(G\) on \(L^{0,q}_2 (E_\Lambda)_d\) decomposes into a finite sum of discrete classes. The Laplacian \(\square\) has only finitely many eigenvalues on \(L^{0,q}_2 (E_\Lambda)\) and the sum of the eigenspaces coincides with \(L^{0,q}_2 (E_\Lambda)_d\). We then have a complex \[ \cdots \longrightarrow L^{0,q}_2 (E_\Lambda)_d \xrightarrow{\bar\partial} L^{0,q+1}_2 (E_\Lambda)_d \longrightarrow \cdots \] and the cohomology of this complex at the \(q\)-th stage is isomorphic to \(H^{0,q}_2 (E_\Lambda)\). If \(\varphi\) is a \(K\)-finite \(C^\infty\)-function with compact support on \(G\), \(\varphi\) defines an operator \(T_\varphi^q\) on \(L^{0,q}_2 (E_\Lambda)_d\) and \(\{T_\varphi^q\}\) forms an endomorphism of finite rank of the above complex. So it suffices to calculate the alternating sum of the traces of the operators \(T_\varphi^q\) on \(L^{0,q}_2 (E_\Lambda)_d\). The alternating sum of the traces of \(T_\varphi^q\) is given by an integral over the union of conjugacy classes of a compact Cartan subgroup. The integral involves the character of the representation of \(K\) and the integral is evaluated by using Weyl’s character formula for irreducible representations of \(K\) and coincides with Harish-Chandra’s character formula for the discrete series for \(G\).”

This paper stimulated further research, see R. Hotta [J. Math. Soc. Japan 23, 384–407 (1971; Zbl 0213.13701)], W. Schmid [Ann. Math. (2) 93, 1–42 (1971; Zbl 0291.43013)], R. Parthasarathy [Ann. Math. (2) 96, 1–30 (1972; Zbl 0249.22003)].

Reviewer: J. A. C. Kolk