Geometric realization of the relative discrete series and decomposition of L²-kernels of invariant elliptic operators on homogeneous spaces of unimodular Lie groups

The L²-index theorem of Connes and Moscovici (1982) computes, for a connected unimodular Lie group $G$ and a $G$-invariant elliptic operator $D$ on a homogeneous space $G/H$, the real-valued $G$-index $\mathrm{ind}_G(D)$ from the principal symbol, and refines it to a class in $K_0(C_r^*(G))$. The paper left open several issues: the integrality of the $G$-index, the computation of $K_0(C_r^*(G))$ together with an intrinsic formula for the refined index, the vanishing of the 'wrong' kernel of the invariant Dirac-type operators $A_\lambda^-$ on nilpotent groups at small central parameters, and the decomposition of $L^2$-kernels into discrete-series representations beyond certain structural classes of groups. The K-theoretic components of this list are resolved (Connes–Kasparov conjecture for almost connected groups: Chabert–Echterhoff–Nest 2003; explicit formulas via higher orbital integrals for reductive groups), while the representation-theoretic core remains open: whether $\ker A_\lambda^-=\{0\}$ for all central parameters (and, in general form, Kasparov's 1983 Dirac-realization conjecture for square-integrable representations), and whether the $L^2$-kernel of every invariant elliptic operator on $G/H$ is a finite direct sum of irreducible square-integrable representations for general connected unimodular type-I groups $G$.

Background

Let $G$ be a connected unimodular Lie group (unimodular means a left-invariant Haar measure is also right-invariant) and $H\subset G$ a compact subgroup. A $G$-invariant elliptic differential or pseudo-differential operator $D$ acting between sections of homogeneous vector bundles over $M=G/H$ is typically non-Fredholm on $L^2$, because $M$ is noncompact; however, $D$ commutes with the natural unitary representation of $G$ on $L^2(M)$, and this symmetry restores an index theory. The algebra of bounded operators on $L^2(M)$ commuting with the $G$-action (the commutant of the representation) carries a canonical trace, and the von Neumann dimension $\dim_G V$ (the $G$-dimension) of a $G$-invariant closed subspace $V\subset L^2(M)$ is the value of this trace on the orthogonal projection onto $V$. The real-valued $G$-index is $\mathrm{ind}_G(D)=\dim_G\ker D-\dim_G\ker D^*$. Connes and Moscovici (1982) proved an Atiyah–Singer-type formula computing $\mathrm{ind}_G(D)$ from the principal symbol, with characteristic classes expressed through the relative Lie algebra cohomology of the pair $(\mathfrak g,\mathfrak h)$ — the cohomology of the complex of $G$-invariant differential forms on $G$ that are horizontal for the $H$-action — reducing the computation to the longitudinal index theorem of a suitable compact foliation (the analogue of the Atiyah–Singer theorem for operators that differentiate only along the leaves of a foliation) and thereby avoiding any hypothesis that $G$ contain a lattice. Two structural features of this theorem drive the problems considered here. First, $\mathrm{ind}_G(D)$ is a real number depending on the normalization of Haar measure on $G$; the paper already produces, in the nilpotent setting, invariant operators whose indices have irrational ratios, so no choice of Haar measure makes all $G$-indices integral. The proposed remedy is to refine the index to a class $\mathrm{Ind}_G$ in $K_0(C_r^*(G))$, the Grothendieck group of finitely generated projective modules over the reduced group $C^*$-algebra $C_r^*(G)$ (the $C^*$-algebra generated by the left regular representation of $G$), a discrete abelian group which pairs with the canonical trace to recover $\mathrm{ind}_G$. Second, the $L^2$-kernel of $D$ is a unitary representation of $G$, and the paper asks when it is built from the discrete series: irreducible unitary representations that are square-integrable modulo the center of $G$ (often called the relative discrete series for non-semisimple groups), each carrying a finite formal degree, i.e. a Plancherel density. This is the noncompact analogue of the Borel–Weil–Bott realization (the classical realization of the irreducible representations of a compact Lie group as spaces of holomorphic sections of line bundles over suitable complex quotient manifolds of the group); for semisimple groups it had been achieved by Atiyah and Schmid via Dirac operators on $G/K$, and for nilpotent groups by Moscovici and Verona through harmonically induced representations (realizations of induced representations on spaces of harmonic forms). Connes and Moscovici proved that, for $G$ semisimple with finite center, and for $G=A\rtimes S$ a semidirect product of a $\mathcal K$-group $A$ with compact center (a connected Lie group admitting a linear functional on its Lie algebra whose coadjoint orbit is contained in an affine subspace parallel to the subspace of functionals vanishing on the Lie algebra of its center) by a reductive group $S$ with compact center, the $L^2$-kernel of every $G$-invariant elliptic operator on $G/H$ is a finite direct sum of irreducible square-integrable representations. For nilpotent $G$ with square-integrable representations they introduced invariant Dirac-type operators $A_\lambda^\pm$ on $G/Z$, the two chiral halves of a Dirac operator twisted by the central character obtained by exponentiating $\lambda$, where $Z$ is the center and $\lambda$ is a linear functional on the Lie algebra whose coadjoint isotropy group — the subgroup of $G$ fixing $\lambda$ under the natural action of $G$ on the dual of its Lie algebra — is exactly $Z$ (by the Kirillov orbit method, which parametrizes the irreducible unitary representations of a nilpotent Lie group by the coadjoint orbits on the dual of its Lie algebra, such $\lambda$ parametrize precisely the square-integrable irreducibles $\pi_\lambda$), and proved that the multiplicities $m_\lambda^\pm$ of $\pi_\lambda$ in $\ker A_\lambda^\pm$ satisfy $m_\lambda^+-m_\lambda^-=1$, while the 'wrong' kernel $\ker A_p^-$ vanishes for all sufficiently large integral values of the central parameter $p$. The subsequent history of the problem list follows the operator-algebraic refinement. Kasparov (1983), independently, constructed the $K$-theoretic index for invariant elliptic operators and formulated the conjecture that every square-integrable irreducible representation of a connected unimodular Lie group is realized on the $L^2$-kernel of a Dirac operator on $G/G_K$ ($G_K$ a maximal compact subgroup) with one of the two half-kernels vanishing. The computation of $K_0(C_r^*(G))$ became the Connes–Kasparov conjecture — Dirac induction from the representation ring of $K$ being an isomorphism onto $K_*(C_r^*(G))$ — proved for all locally compact groups with cocompact connected component by Chabert, Echterhoff and Nest (2003), as a corollary of the Baum–Connes assembly map isomorphism. The $L^2$-index formula was extended to proper cocompact actions of unimodular groups on arbitrary manifolds by Wang (2014), and for real reductive groups Hochs, Song and Tang (2021) proved an index theorem for the pairings of $K_*(C_r^*(G))$ with explicitly constructed cyclic cocycles (higher orbital integrals) that determine the equivariant index completely; Clare, Higson and Román (2025) characterized the Connes–Kasparov assembly map through a Mackey embedding of motion-group $C^*$-algebras. In contrast, the two representation-theoretic questions — the exact Dirac realization of the discrete series at small parameters, and the decomposition of $L^2$-kernels beyond the structural classes above — were advanced only by Chern (1997) and have remained open since.

Problem Statement

The problem is the open core of the issues posed at the end of Connes–Moscovici (1982) concerning $L^2$-kernels of $G$-invariant elliptic operators on homogeneous spaces of unimodular Lie groups; the other issues from the same list — the computation of $K_0(C_r^*(G))$ and the intrinsic $K$-theoretic index formula — were subsequently resolved for all Lie groups with finitely many components (Chabert–Echterhoff–Nest 2003, reference 4). The common objective of the two linked questions below is the exact geometric realization of the relative discrete series by $L^2$-kernels of invariant elliptic operators, for general unimodular Lie groups. (P1) Small-parameter vanishing and exact realization. Let $G$ be a connected, simply connected nilpotent Lie group with center $Z$ admitting square-integrable irreducible representations modulo $Z$; let $\lambda$ be a linear functional on the Lie algebra of $G$ whose coadjoint isotropy group is $Z$; and let $A_\lambda^\pm$ be the invariant Dirac-type operators on $G/Z$ twisted by the central character associated with $\lambda$ (the character of $Z$ obtained by exponentiating $\lambda$), in the sense of Connes–Moscovici. Determine whether $\ker A_\lambda^-=\{0\}$ for every such $\lambda$ — equivalently, whether the square-integrable representation $\pi_\lambda$ is realized exactly, with multiplicity one, as $\ker A_\lambda^+$ — or produce a $\lambda$ for which $\ker A_\lambda^-\neq\{0\}$. The known results are $m_\lambda^+-m_\lambda^-=1$ and the vanishing of $\ker A_p^-$ only for all sufficiently large central parameters $p$. As an alternative general form of the same question, one may instead prove or refute Kasparov's 1983 conjecture: is every irreducible square-integrable representation of a connected unimodular Lie group $G$ realized on the $L^2$-kernel of a Dirac operator on $G/G_K$, where $G_K$ is a maximal compact subgroup, with either $\ker D_\pi^-=\{0\}$ or $\ker D_\pi^+=\{0\}$? The general Kasparov form and the nilpotent form present the same question; (P2) is an independent second question. (P2) Decomposition of $L^2$-kernels in general. Let $G$ be a connected unimodular Lie group of type I (every factor representation generates a type I von Neumann algebra, so the regular representation decomposes over irreducibles) which possesses square-integrable representations; let $H\subset G$ be compact; and let $D$ be a $G$-invariant elliptic operator on $G/H$. Determine whether the $L^2$-kernel of $D$ is always a finite direct sum of irreducible square-integrable representations of $G$. This is known for $G$ semisimple with finite center and for $G$ a semidirect product of a $\mathcal K$-group with compact center by a reductive group with compact center, and for related semidirect-product classes; the question is open for general such $G$, including the residual central and compact extensions allowed by the structure theory of groups with square-integrable representations.

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