Blowups, deformations to the normal cone and groupoids

• Georges Skandalis

Room: Amphi Schwartz IMT building 1R3 We give a systematic construction of some deformation groupoids which recover many previous ones. Our construction gives rise to several extensions. We compute the corresponding K-theory maps. This is joint work with Claire Debord.

A Hilbert bundle description of differential K-theory

• John Lott

Room: Amphi Schwartz IMT building 1R3 We give an infinite-dimensional description of the differential K-theory of a manifold. The construction uses superconnections on Hilbert bundles and eta forms. We describe the pushforward of a finite-dimensional cycle under a proper submersion with a Riemannian structure. Finally, we give a model for twisted differential K-theory. This is joint work with Alexander Gorokhovsky

K-types of tempered representations

• Peter Hochs

Room: Amphi Schwartz IMT building 1R3

Fixed Point Theorem and Character Formula

• Hang Wang

Room: Amphi Schwartz IMT building 1R3 This talk is motivated by the bridge between geometry and representation. Weyl character formula, which describes characters of irreducible representations of a compact Lie group, can be obtained geometrically from the Atiyah-Segal-Singer or the Atiyah-Bott fixed point theorem. We present a fixed point theorem associated to a proper action of a Lie group on a manifold. We obtain Harish-Chandra character formula, the noncompact analogue of the Weyl Character formula, geometrically from the fixed point theorem. This is joint work with Peter Hochs.

Heat kernel methods and algebraic index theorem

• Rudy Rodsphon

Room: Amphi Schwartz IMT building 1R3 One way to describe succinctly local index theory on closed spin manifolds could be the following slogan of Quillen : Dirac operators are a "quantization" of connections, and index theory is a "quantization" of the Chern character. As is well-known, this leads to proofs of the index theorem using heat kernel methods. For non necessarily spin manifolds, pseudodifferential operators and their symbolic calculus play a crucial role in the original proofs of the index theorem. However, symbols may also be viewed as a deformation quantization of functions on the cotangent bundle, which has led to other fruitful approaches to index theory through a "quantization" process. Methods used in these two different quantization pictures do not seem to be quite related a priori. Based on ideas of Perrot, the upshot of the talk will be that it is possible to implement an algebraic version of the heat kernel method in the deformation quantization picture, which has many avantages over the original one. In particular, we recover Nest-Tsygan's algebraic index theorem for a certain class of symplectic manifolds in a very natural way.

The analytic index of longitudinal elliptic operators on an open foliated manifold

• Xiang Tang

Room: Amphi Schwartz IMT building 1R3 In this talk, we will introduce the concept of Roe C\*-algebra for a locally compact groupoid whose unit space is in general not compact, and that is equipped with an appropriate coarse structure and Haar system. Using Connes' tangent groupoid method, we will define an analytic index for an elliptic differential operator on a Lie groupoid equipped with additional metric structure, which takes values in the K-theory of the Roe C\*-algebra. And we will discuss applications of our developments to longitudinal elliptic operators on an open foliated manifold. This is joint work with Rufus Willett and Yi-Jun Yao.

Witten's perturbation on strata with general adapted metrics

• Jesus Alvarez Lopez

Room: Amphi Schwartz IMT building 1R3 The lecture is about a version of the Morse inequalities that we have proved on strata. The proof uses the minimum and maximum ideal boundary conditions of the Witten's perturbation of the de Rham complex, with respect to what we call a relative Morse function and a general adapted metric. This can be considered as a result about intersection cohomology with arbitrary perversities. All details of the proof were recently finished, correcting some computations of a previous version. This is a joint work with Manuel Calaza and Carlos Franco.

The logarithmic index of vector fields and the theory of residue

• Alexandre Aleksandrov

Room: Amphi Schwartz IMT building 1R3 We introduce the notion of logarithmic index of vector fields and differential forms (not necessarily regular) given on singular varieties. Then the corresponding theory will be developed in various settings and some useful relations with classical theories of index and residue will be discussed. Our approach is mainly based on the theory of residues of meromorphic differential forms logarithmic along subvarieties with arbitrary singularities developed by the author in the past few years. As illustrations, we also discuss elementary methods for computing the index on Cohen-Macaulay curves, complete intersections, normal and determinantal varieties, and others.

Calculating the Fredholm index on Lie manifolds

• Karsten Bohlen

Room: Amphi Schwartz IMT building 1R3 We review Fredholm conditions for operators contained in the pseudodifferential calculus for manifolds with a Lie structure at infinity. Then we study the problem of obtaining Atiyah-Singer type index formulas for Fredholm operators contained in the calculus. Time permitting we discuss some of the applications of the results.

Local linear forms on pseudodifferential operators and index theory

• Sylvie Paycha

Room: Amphi Schwartz IMT building 1R3 This talk discusses **local** linear forms $\Lambda: A\mapsto \Lambda(A)$ on classical pseudo-differential operators on a closed manifold, namely linear forms of the type $\Lambda(A)=\int_M\lambda_A(x)\, dx$ given by a density $\lambda_A(x)\, dx$ on the manifold $M$ and their relevance in index theory. Local linear forms are spanned by the well-known Wodzicki residue $A\longmapsto {\rm Res}(A)$ on integer order operators and the somewhat lesser known canonical trace $A\longmapsto {\rm TR} (A)$ on non-integer order operators (joint work with S. Azzali). For a **holomorphic perturbation** $A(z)$ of a differential operator $A(0)=A$, these two linear forms relate by (joint work with S. Scott) \[ (1)\,\,\,\,\, \,\,lim_{z\to 0}TR(A(z))= -\frac{1}{2}\, Res(A^\prime(0)), \] where the residue has been extended to logarithms. Inspired by Gilkey's approach using invariance theory, for a family $A(z)$ of geometric operators, we showed (joint work with J. Mickelsson) that the density $ {\rm res}_x(A^\prime (0))$ arising in the r.h.s. of (1) is an invariant polynomial which can be expressed in terms of Pontryagin forms on the tangent bundle and Chern forms on the auxillary bundle. A $\mathbb{Z}_2$-graded generalisation of (1) applied to an appropriate holomorphic perturbation of the identity built from a Dirac operator $D=D_+\oplus D_-$ acting on a $\mathbb{Z}_2$-graded vector bundle, expresses the **index** of $D_+$ in terms of a Wodzicki residue. As a result of their locality, the canonical trace and the Wodzicki residue are preserved under lifting to the universal covering of a closed manifold; consequently formula (1) lifts to coverings. This lifted analoque of (1) yields an expression of the $L^2$-index of a lifted Dirac operator in terms of the Wodzicki residue of the logarithm of its square (joint work with S. Azzali).

Pairings for pseudodifferential symbols

• Alexander Gorokhovsky

Room: Amphi Schwartz IMT building 1R3 We compare different constructions of cyclic cocycles for the algebra of complete symbols of pseudodifferential operators. Our comparison result leads to index-theoretic consequences and a construction of invariants of the algebraic $K$-theory of the algebra of pseudodifferential symbols. This is a joint work with H. Moscovici.

Elliptic Operators Associated with Groups of Quantized Canonical Transformations

• Elmar Schrohe

Room: Amphi Schwartz IMT building 1R3 Given a Lie group $G$ of quantized canonical transformations acting on the space $L^2(M)$ over a closed manifold $M$, we define an algebra of so-called $G$-operators on $L^2(M)$. We show that to $G$-operators we can associate symbols in appropriate crossed products with $G$, introduce a notion of ellipticity and prove the Fredholm property for elliptic elements. This framework encompasses many known elliptic theories, for instance, shift operators associated with group actions on $M$, transversal elliptic theory, transversally elliptic pseudodifferential operators on foliations, and Fourier integral operators associated with coisotropic submanifolds.

On the work of Boris Sternin in elliptic theory

• Anton Savin

Room: Amphi Schwartz IMT building 1R3 In this talk, I intend to give a survey of the main results obtained by the late Professor Boris Sternin (1939--2017) and his collaborators in the theory of elliptic operators. His research embraced a variety of topics such as Sobolev problems (relative elliptic theory), elliptic theory on manifolds with singularities (including surgery techniques to calculate indices and K-homology techniques to obtain homotopy classification of elliptic operators), and noncommutative elliptic theory for operators associated with group actions (including the uniformization method, which allows one to compute the indices of elliptic operators).

Lefschetz trace formulas for flows on foliated manifolds

• Yuri Kordyukov

Room: Amphi Schwartz IMT building 1R3 In this talk, we will discuss Lefschetz trace formulas for foliated flows on compact manifolds equipped with a codimension one foliation. Our main motivation comes from Deninger's approach to the study of arithmetic zeta-functions. First, we will recall such a formula, due to Alvarez Lopez and the speaker, in the case when the flow has no fixed points and its orbits are everywhere tranverse to the leaves of the foliation. Then we will consider the case when the flow may have fixed points and describe an approach to Lefschetz trace formulas based on pseudodifferential b-calculus developed by Melrose. We will report on the recent progress in this direction. This is joint work with Jesus Alvarez Lopez and Eric Leichtnam.

Elements "Gamma" finiment sommables pour les groupes hyperboliques.

• Michael Puschnigg

Room: Amphi Schwartz IMT building 1R3 On donne une construction uniforme d'un module de Fredholm finiment sommable qui représente l’élément "Gamma" dans $KK_G(C,C)$ d'un groupe hyperbolique G. On obtient une estimation explicite de son degré de sommabilité en termes de la cardinalité d'un ensemble fini symétrique de générateurs S de G et de la constante de hyperbolicité $\delta$ du graphe de Cayley de (G,S). (Travail commun avec J.M.Cabrera).

Noncommutative geometry, conformal geometry, and the cyclic homology of crossed-product algebras.

• Raphael Ponge

Room: Amphi Schwartz IMT building 1R3 In the first part of the talk, I will present some joint work with Hang Wang. We use noncommutative geometry to obtain a local index formula in conformal geometry that takes into account the action of an arbitrary group of conformal diffeomorphisms. This leads us to a construction of a whole new family of conformal invariants. The computation of these invariants uses the explicit computation of the cyclic homology of crossed-product algebras by means of explicit quasi-isomorphisms that I constructed recently. This will be the topic of the 2nd part of the talk. The results are expressed in terms of suitable versions of equivariant (co)homology. As a result this allows us to compute the conformal invariants of the 1st part in terms of equivariant characteristic classes.

Proper actions of Lie groups and higher APS index theory

• Hessel Posthuma

Room: Amphi Schwartz IMT building 1R3 In this talk I’ll report on joint work in progress with Paolo Piazza about higher APS index theory in the presence of a Lie group symmetry. I will first review the higher index theorem for proper, co-compact Lie group actions on closed manifolds. After that I will consider the generalization to the APS-setting where the manifold has a boundary and is equipped with a Dirac operator which is invariant under the action of the group. The higher indices of this operator are associated to smooth group cocycles and defined via the pairing of (relative) K-theory with (relative) cyclic cohomology. Comparing with the case of closed manifolds, I will explain how the APS setting differs from it and requires a much more involved analysis.

The asymptotics of the holomorphic torsion forms

• Martin Puchol

Room: Amphi Schwartz IMT building 1R3 The holomorphic torsion is a spectral invariant defined by Ray and Singer. Bismut and Vasserot have computed its asymptotic behavior when it is associated with growing tensor power of a positive line bundle. Then they extended their result when these powers are replaced by symmetric powers of a positive bundle of arbitrary rank. These formulas have played a role in Arakelov geometry. The holomorphic torsion has a generalization in the family setting: the holomorphic torsion forms. In this talk, we will extend Bismut-Vasserot's work and present an asymptotic formula for the torsion forms associated with the direct image of $L^{\otimes p}$, where $L$ is a line bundle satisfying a positivity assumption along the fibers. A key step for this is to use of the Toeplitz operators.

Lie groupoids, complete metrics and secondary invariants on stratified manifolds

• Vito Felice Zenobi

Room: Amphi Schwartz IMT building 1R3

Fredholm conditions on non-compact manifolds through groupoids

• Catarina Carvalho

Room: Amphi Schwartz IMT building 1R3 In many classes of non-compact manifolds, a (pseudo)differential operator is Fredholm if, and only if, it is elliptic and a certain family of invariant operators is invertible. In this talk, we discuss this type of Fredholm conditions in the framework of Lie groupoids over manifolds with corners, and provide a setting where they can be explicitly identified. Representation theory of groupoid $C^*$-algebras plays a significant role, namely recent work by Roch and Nistor, Prudhon on strictly spectral and exhaustive families. We discuss examples, and consider, in particular, the commutative case, where we see that the classical Atiyah-Singer index formula applies. As a consequence, we obtain an index formula for even-dimensional Callias type operators with unbounded potentials. This is joint work with V. Nistor and Y. Qiao.

A groupoid approach to pseudodifferential operators

• Robert Yuncken

Room: Amphi Schwartz IMT building 1R3 The tangent groupoid is a geometric device for glueing a pseudodifferential operator to its principal symbol via a deformation family. We will discuss a converse to this: briefly, pseudodifferential kernels are precisely those distributions that extend to distributions on the tangent groupoid that are essentially homogeneous with respect to the natural R+-action. One could see this as a simple new definition of a classical pseudodifferential operator. Moreover, we will show that, armed with an appropriate generalization of the tangent groupoid, this approach allows us to easily construct more exotic pseudodifferential calculi, such as the Heisenberg calculus. (Joint work with Erik van Erp.)

K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS

• Paul Baum

Room: Amphi Schwartz IMT building 1R3 K-homology is the dual theory to K-theory. The BD (Baum-Douglas) isomorphism of Atiyah-Kasparov K-homology and K-cycle K-homology provides a framework within which the Atiyah-Singer index theorem can be extended to certain differential operators which are hypoelliptic but not elliptic. This talk will consider such a class of differential operators on compact contact manifolds. These operators have been studied by a number of mathematicians. Operators with similar analytic properties have also been studied (e.g. by Alain Connes and Henri Moscovici). Working within the BD framework, the index problem will be solved for these operators. The Connes-Thom isomorphism plays an essential role in the proof. This is joint work with Erik van Erp.

The role of groupoids in the index problem for hypoelliptic operators.

• Erik van Erp

Room: Amphi Schwartz IMT building 1R3 In the 1980s, Alain Connes gave a conceptually appealing proof of the Atiyah-Singer index theorem using the tangent groupoid. Over time, the tangent groupoid was generalized to more complicated analytic settings. I will discuss the role played by groupoids in the resolution of the index problem for hypoelliptic differential operators on contact manifolds.

Stratified surgery and the signature operator.

• Paolo Piazza

Room: Amphi Schwartz IMT building 1R3 In this talk I will explain how a microlocal approach to elliptic theory on (the regular part of) a Witt or, more generally, a Cheeger space allows to extend the mapping-surgery-to-analysis philosophy to these classes of stratified spaces. In particular, I will report on some recent work with Pierre Albin where, building on the analytic theory developed by Albin, Leichtnam, Mazzeo and myself, we map the Browder-Quinn surgery sequence for Witt or Cheeger stratified spaces to suitable K-theory sequences.

Dirac-type operators on stratified spaces

• Pierre Albin

Room: Amphi Schwartz IMT building 1R3 I will describe joint work with Jesse Gell-Redman on the index theorem on stratified spaces.

The Chern conjecture for affine manifolds

• Weiping Zhang

Room: Amphi Schwartz IMT building 1R3 I would like to present the joint work with Huitao Feng on the proof of the Chern conjecture which states that the Euler characteristic of a closed affine manifold equals to zero.

Conical degeneration of geometric invariants

• Xianzhe Dai

Room: Amphi Schwartz IMT building 1R3 Conical singularities occur quite often and naturally. By conical degeneration we mean a family of smooth metrics limiting to a singular metric of conical type. Under rather general conditions, the eigenvalues and eigenfunctions, and heat kernels will converge. It is rather different for the global geometric invariants such as eta invariant and analytic torsion. We will discuss some recent work in this direction.

Bergman kernels on symplectic manifolds and applications.

• Xiaonan Ma

Room: Amphi Schwartz IMT building 1R3 A suitable notion of ``holomorphic section'' of a prequantum line bundle on a compact symplectic manifold is the eigensections of low energy of the Bochner Laplacian acting on high $p$-tensor powers of the prequantum line bundle. We explain the asymptotic expansion of the corresponding kernel of the orthogonal projection as the power p tends to infinity. This implies the compact symplectic manifold can be embedded in the corresponding projective space. With extra effort, we show the Fubini-Study metrics induced by these embeddings converge at speed rate $1/p^{2}$ to the symplectic form. We explain also its implication on Bezerin-Toeplitz quantizations.

On the Laplace-Beltrami operator on compact complex spaces

• Francesco Bei

Room: Amphi Schwartz IMT building 1R3 During the last decades analysis on complex projective varieties endowed with the Fubini-Study metric and more generally on Hermitian complex spaces has been an active research field. In this talk we will present some recent results about the Laplacian in the setting of compact Hermitian complex spaces. More precisely let (X, h) be a compact and irreducible Hermitian complex space of complex dimension v > 1. We will show that the Friedrich extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover for the Friedrich extension of the Laplace-Beltrami operator we will also provide an estimate for the growth of its eigenvalues. Finally we will give some applications to the Hodge-Dolbeault operator in the setting of Hermitian complex spaces of complex dimension 2.

Mapping rough assembly to homology

• Alexander Engel

Room: Amphi Schwartz IMT building 1R3 We will discuss a transformation of the rough assembly map to a map on large scale homology groups. We will then focus on a specific aspect in this transformation, namely the computation of the homology of (a smooth subalgebra of) the uniform Roe algebra.

An index theorem for the Dirac operator on Lorentzian manifolds

• Christian Bar

Room: Amphi Schwartz IMT building 1R3 We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same formal expression as in the index formula of Atiyah-Patodi-Singer for Riemannian manifolds with boundary. If time permits, an application to quantum field theory (the computation of the chiral anomaly) will be sketched. This is the first index theorem for Dirac operators on *Lorentzian* manifolds and, from an analytic perspective, the methods to obtain it are quite different from the classical Riemannian case. This is joint work with Alexander Strohmaier.

Quasi-asymptotically conical Calabi-Yau manifolds

• Frédéric Rochon

Room: Amphi Schwartz IMT building 1R3 We will explain how to construct new examples of quasi-asymptotically conical (QAC) Calabi-Yau manifolds that are not quasi-asymptotically locally Euclidean (QALE). Our strategy consists in introducing a natural compactification of QAC-spaces by manifolds with fibred corners and to give a definition of QAC-metrics in terms of a natural Lie algebra of vector fields on this compactification. Using this and the Fredholm theory of Degeratu-Mazzeo for elliptic operators associated to QAC-metrics, we can in many instances obtain Kahler QAC-metrics having Ricci potential decaying sufficiently fast at infinity. We can then obtain QAC Calabi-Yau metrics in the Kahler classes of these metrics by solving a corresponding complex Monge-Ampere equation. This is a joint work with Ronan Conlon and Anda Degeratu.

Secondary invariants for two-cocycle twists

• Sara Azzali

Room: Amphi Schwartz IMT building 1R3 We consider Dirac operators, on the universal covering of a closed manifold, that are invariant under the projective action associated to a two-cocycle of the fundamental group. These operators give interesting invariants analogous to those studied in $L^2$-index theory for covering spaces, or more generally higher index theory. The key property of this setting is that the twist by a two-cocycle naturally yelds a $C^*$-algebraic bundle of arbitrary small curvature. We will describe the construction of eta and rho invariants, prove an Atiyah–Patodi–Singer index theorem in this setting, and discuss some of their geometric properties. This is based on joint work with Charlotte Wahl.

Noncommutative products of Euclidean spaces

• Giovanni Landi

Room: Amphi Schwartz IMT building 1R3 We present natural families of coordinate algebras of noncommutative products of Euclidean spaces. These coordinate algebras are quadratic ones associated with an R-matrix which is involutive and satisfies the Yang-Baxter equations. As a consequence they enjoy a list of nice properties, being regular of finite global dimension. Notably, we have eight-dimensional noncommutative euclidean spaces which are particularly well behaved and are parametrised by a two-dimensional sphere. Quotients include noncommutative seven-spheres as well as noncommutative "quaternionic tori". There is invariance for an action of $SU(2) \times SU(2)$ in parallel with the action of $U(1) \times U(1)$ on a "complex" noncommutative torus which allows one to construct quaternionic toric noncommutative manifolds. Additional classes of solutions are disjoint from the classical case.

A Baum-Connes conjecture for singular foliations

• Iakovos Androulidakis

Room: Amphi Schwartz IMT building 1R3 A large class of manifolds M endowed with a Stefan-Sussmann singular foliation, admit a decomposition which records the "height" of the singularities involved. These are the manifolds whose singular foliation is defined from a Lie groupoid, and the singularity height can be formulated using dimension. Using this decomposition we can formulate an assembly map for singular foliations as such, which is an isomorphism under suitable amenability conditions. The assembly map allows to calculate the K-theory of the foliation C*-algebra. This is joint work with Georges Skandalis.

Roe algebras, coarse geometry, and exactness.

• Rufus Willett

Room: Amphi Schwartz IMT building 1R3 Roe algebras are C*-algebras associated to possibly open manifolds, or more general metric spaces; they are invariants of the large-scale geometry. Roe algebras are motivated by the fact that (nice enough, e.g. Dirac type) differential operators on a Riemannian manifold have higher indices in the K-theory of the Roe algebra; if the manifold happens to be closed, the Roe algebra is just the compact operators and one recovers the classical integer index this way. For ‘good' spaces, the Roe algebra remembers essentially all the large-scale geometry of the space X, while in ‘bad’ ones, analytic difficulties arise leading to counterexamples to Baum-Connes type conjectures. I’ll try to survey what makes a space ‘good’ versus ‘bad’ in this context, and some of the consequences. Parts of this talk will be based on joint work with several people: Paul Baum, Erik Guentner, John Roe, Jan Spakula, Stuart White, and Guoliang Yu.

Spectral triples and non-commutative fractals

• Daniele Guido

Room: Amphi Schwartz IMT building 1R3 Self-similar nested fractals are studied from a functional point of view, and this provides a way to quantize them, namely to produce a self-similar noncommutative C*-algebra containing the continuous functions on the fractal as a sub-algebra. For the noncommutative C\*-algebra associated with the Sierpinski gasket, the representations are studied, it is shown that a noncommutative Dirichlet form can be defined, which restricts to the classical energy form on the gasket, and a spectral triple is proposed. Such triple reconstructs in particular the Dirichlet form via the formula $a\to res_{s=\delta}\ tr(|D|^{-s/2}|[D,a]|^2 |D|^{-s/2})$, for a suitable $\delta$. Work in progress with F.Cipriani, T.Isola and J-L.Sauvageot.

Hilsum bordisms and unbounded KK-theory

• Robin Deeley

Room: Amphi Schwartz IMT building 1R3 Inspired by naturally occurring geometric examples, Baaj and Julg defined the notion of an unbounded cycle in KK-theory. Likewise, based on operators associated to manifolds with boundary, Hilsum defined the notion of a bordism in the context of unbounded KK-theory. I will discuss joint work with Magnus Goffeng and Bram Mesland in which, we defined an abelian group that is essentially unbounded KK-cycles modulo Hilsum's notion of bordism. This group maps to the standard Kasparov group via the bounded transform and in the commutative case can be related to the geometric model for K-homology due to Baum and Douglas.

Integrable lifts for transitive Lie algebroids

• Paolo Antonini

Room: Amphi Schwartz IMT building 1R3 In this seminar we report on work in progress with I. Androulidakis and I. Marcut In many constructions in noncommutative geometry, the passage from a singular space to a C* algebra involves a Lie groupoid as an intermediate desingularization space. The infinitesimal datum of a Lie groupoid is a Lie algebroid and they appear independently, for instance in : -theory of foliations -Poisson geometry -Gauge theory. However in general is not possible to integrate a Lie algebroid to a Lie groupoid ( in contrast to the theory of Lie algebras). Firstly we will be concerned with the discussion of Lie algebroids: basic definitions, examples, the integration problem, the obstructions to the integrability of Crainic-Fernandes and the discussion of the first non integrable example given by Molino. Then we will explain our idea of "removing" the obstructions of a transitive algebroid, passing to a suitable integrable extension. In these cases one can use this integrable lift to perform some of the basic constructions of index theory and noncommutative geometry.

On the Parthasarathy formula for quantized irreducible flag manifolds

• Marco Matassa

Room: Amphi Schwartz IMT building 1R3 The Parthasarathy formula expresses the square of the Dirac operator on a symmetric space in terms of central elements of the corresponding enveloping algebra. We investigate whether a result of this type also holds for quantized irreducible flag manifolds, using the Dolbeault-Dirac operators introduced by Krähmer and Tucker-Simmons. We show that a Parthasarathy-type formula requires certain quadratic commutation relations in the quantum Clifford algebra defined by the named authors. For quantum projective spaces these relations hold, and we obtain a result which is as close as possible to the classical case. On the other hand this is not the case for all other irreducible flag manifolds.