arXiv subjects

Dan Ciubotaru

Publications and source records attributed to Dan Ciubotaru.

At least 19 recordsLinked to original sources

arXiv2026

Jantzen filtrations for graded affine Hecke algebras

We prove the Jantzen conjecture for standard modules of graded affine Hecke algebras with equal parameters, real central character in the dominant deformation direction. The Jantzen layers are shown to be semisimple, and the multiplicities are given by the full local intersection cohomology of the corresponding orbit closures. The proof identifies the algebraic standard-to-contragredient map with the costalk-to-stalk map on a contracting transverse slice. Hard Lefschetz theorem determines its elementary divisors, and a residual grading on the specialized convolution algebra proves semisimplicity. An example in type $A$ is included to show that dominance is necessary.

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arXiv2026

Symplectic Dirac operators on homogeneous spaces

We define symplectic Dirac operators on homogeneous spaces and study their representation-theoretic role. For an invariant polarization, the symplectic Dirac operator decomposes into two symplectic Dolbeault operators. We compute their commutator as the natural symplectic analogue of the square of the classical Dirac operator. Our first main result gives a necessary and sufficient condition for this commutator to satisfy a Parthasarathy-type formula. We further prove that, whenever this condition fails, no cubic perturbation of the symplectic Dolbeault operators can yield such a formula, in contrast with Kostant's cubic Dirac operator in the orthogonal setting. As applications, we establish an ${\mathfrak s}{\mathfrak l}_2$-structure generated by the symplectic Dolbeault operators and derive Dirac-type inequalities for unitary representations of Hermitian symmetric spaces labelled by the levels of the symmetric powers of the antiholomorphic tangent space at the identity. The level-zero inequality recovers the standard Parthasarathy-Dirac inequality, while the higher levels inequalities yield new constraints. For $SU(1,n)$, we show that, for representations with a specific Kraljević corner, the level-one inequality strengthens all basic Parthasarathy inequalities of the first kind for particular $K$-types, precisely those satisfying an explicit highest-weight condition.

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arXiv2026

Generic spherical unitary dual for Chevalley groups over local fields

We prove that a generic spherical parameter for the graded affine Hecke algebra with equal parameters is unitary if and only if its normalized intertwining forms are positive on the reflection representation and on the irreducible constituents of its second symmetric power. The proof combines a signature formula for the reflection representation, Yu's uniform two-wall operation, a simply-laced Jantzen recursion, and a root-poset analysis. In types B_n and C_n, we prove more sharply that the reflection representation V and the traceless-diagonal constituent of Sym^2(V) suffice. We also give a shorter second proof when arbitrary Weyl-group types are allowed. The consequence of the first proof (via Barbasch-Vogan petite K-types) is that the generic spherical unitary dual of a Chevalley group over the real numbers or a nonarchimedean local field, and for simply-laced groups and complex symplectic groups also over the complex numbers, is independent of the field. Moreover, the answer has a simple uniform description as the Weyl conjugates of a disjoint union of a power of 2 alcoves in the affine Weyl group arrangement in the fundamental Weyl chamber, where the exponent is the matching number of the Dynkin diagram of the subsystem of short coroots.

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arXiv2026

A nonabelian Fourier transform for tempered unipotent representations, II

We extend the construction of the nonabelian Fourier transform proposed in our previous paper from the space of unipotent elliptic characters to the space of unipotent compact characters of a reductive $p$-adic group, and we investigate its relation with the branching to maximal compact open subgroups. We prove that this Fourier transform is compatible with parahoric restriction for certain groups, including $\mathrm{SL}_n, \mathrm{GL}_n$, and $G_2$. We also relate the compact setting to the elliptic setting using parabolic induction.

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arXiv2026

Endoscopic transfer and the wavefront upper bound conjecture

We verify the local analogue of Jiang's conjecture for the upper bound of the geometric wavefront sets of Arthur type representations of split classical $p$-adic groups with $p\gg 0$, under a certain condition. As a consequence, we also obtain the upper bound conjecture of Kim and the second author, and Hazeltine--Liu--Lo--Shahidi, under the same assumptions. The proof uses Waldspurger's work on the endoscopic transfer supplemented by results of Konno and Varma, as well as the wavefront set computations in the unipotent case by Mason-Brown--Okada and the second author.

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arXiv2025

Unitary representations attached to parabolic subgroups: the case of abelian unipotent radical

We classify the unitary representations with integral infinitesimal character in Lusztig's category of unipotent representations in the case when the geometric parameter space comes from the action of a Levi subgroup on the abelian nilradical of a (maximal) parabolic subalgebra. We organise the unitary representations into microlocal Arthur packets. This is a test case for investigating a conjectural description of unitary representations with integral infinitesimal character.

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arXiv2025

The wavefront set: bounds for the Langlands parameter

For an irreducible smooth representation of a connected reductive $p$-adic group, two important associated invariants are the wavefront set and the (partly conjectural) Langlands parameter. While a wavefront set consists of $p$-adic nilpotent orbits, one constituent of the Langlands parameter is a complex nilpotent orbit in the dual Lie algebra. For unipotent representations in the sense of Lusztig, the corresponding nilpotent orbits on the two sides are related via the Lusztig--Spaltenstein duality, by the work of Ciubotaru--Mason-Brown--Okada. In this paper, we formulate a general upper-bound conjecture and several variants relating the nilpotent orbits that appear in the wavefront set and in the Langlands parameter. We also verify these expectations in some cases, including the depth-zero supercuspidal representations of classical groups and all the irreducible representations of $G_2$.

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arXiv2025

Local character expansions and asymptotic cones over finite fields

We generalise Gelfand-Graev characters to $\mathbb R/\mathbb Z$-graded Lie algebras and lift them to produce new test functions to probe the local character expansion in positive depth. We show that these test functions are well adapted to compute the leading terms of the local character expansion and relate their determination to the asymptotic cone of elements in $\mathbb Z/n$-graded Lie algebras. As an illustration, we compute the geometric wave front set of certain toral supercuspidal representations in a straightforward manner.

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arXiv2024

The ($Γ$-asymptotic) wavefront sets: $GL_n$

Let $G$ be a connected reductive $p$-adic group. As verified for unipotent representations, it is expected that there is a close relation between the (Harish-Chandra-Howe) wavefronts sets of irreducible smooth representations and their Langlands parameters in the local Langlands correspondence via the Lusztig-Spaltenstein duality and the Aubert-Zelevinsky duality. In this paper, we define the $Γ$-asymptotic wavefront sets generalizing the notion of wavefront sets via the $Γ$-asymptotic expansions (in the sense of Kim-Murnaghan), and then study the their relation with the Langlands parameters. When $G=GL_n$, it turns out that this reduces to the corresponding relation of unipotent representations of the appropriate twisted Levi subgroups via Hecke algebra isomorphisms. For unipotent representations of $GL_n$, we also describe the Harish-Chandra-Howe (HCH) local character expansions of irreducible smooth representations using Kazhdan-Lusztig theory, and give another computation of the coefficients in the HCH expansion and the wavefront sets.

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arXiv2024

A nonabelian Fourier transform for tempered unipotent representations

We define an involution on the space of compact tempered unipotent representations of inner twists of a split simple $p$-adic group $G$ and investigate its behaviour with respect to restrictions to reductive quotients of maximal compact open subgroups. In particular, we formulate a precise conjecture about the relation with a version of Lusztig's nonabelian Fourier transform on the space of unipotent representations of the (possibly disconnected) reductive quotients of maximal compact subgroups. We give evidence of the conjecture, including proofs for $\mathsf{SL}_n$ and $\mathsf{PGL}_n$.

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arXiv2023

On the generalized Ramanujan and Arthur conjectures over function fields

Let $G$ be a simple group over a global function field $K$, and let $π$ be a cuspidal automorphic representation of $G$. Suppose $K$ has two places $u$ and $v$ (satisfying a mild restriction on the residue field cardinality), at which the group $G$ is quasi-split, such that $π_u$ is tempered and $π_v$ is unramified and generic. We prove that $π_w$ is tempered at all unramified places $K_w$ at which $G$ is unramified quasi-split. More generally, the set of unitary spherical representations is partitioned according to nilpotent conjugacy classes in the Lie algebra of $G$. We show that if $π_v$ is in the set corresponding to the nilpotent class $N$, and if $π_u$ satisfies an analogous hypothesis, then $π_w$ belongs to the same class $N$, where $w$ is as above. These results are consistent with conjectures of Shahidi and Arthur. The proofs use the Galois parametrization of cuspidal representations due to V. Lafforgue to relate the local Satake parameters of $π$ to Deligne's theory of Frobenius weights. The main observation is that, in view of the classification of unitary spherical representations, due to Barbasch and the first-named author, the theory of weights excludes almost all complementary series as possible local components of $π$. This in turn determines the local Frobenius weights at all unramified places. In order to apply this observation in practice we need a result of the second-named author with Gan and Sawin on the weights of discrete series representations.

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arXiv2023

Wavefront Sets of Unipotent Representations of Reductive $p$-adic Groups II

The wavefront set is a fundamental invariant of an admissible representation arising from the Harish-Chandra-Howe local character expansion. In this paper, we give a precise formula for the wavefront set of an irreducible representation of real infinitesimal character in Lusztig's category of unipotent representations in terms of the Deligne-Langlands-Lusztig correspondence. Our formula generalizes the main result of arXiv:2112.14354, where this formula was obtained in the Iwahori-spherical case. We deduce that for any irreducible unipotent representation with real infinitesimal character, the algebraic wavefront set is a singleton, verifying a conjecture of M\oeglin and Waldspurger. In the process, we establish new properties of the generalized Springer correspondence in relation to Lusztig's families of unipotent representations of finite reductive groups.

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arXiv2023

Wavefront Sets of Unipotent Representations of Reductive $p$-adic Groups I

The wavefront set is a fundamental invariant arising from the Harish-Chandra-Howe local character expansion of an admissible representation. We prove a precise formula for the wavefront set of an irreducible Iwahori-spherical representation with `real infinitesimal character' and determine a lower bound for this invariant in terms of the Deligne-Langlands-Lusztig parameters. In particular, for the Iwahori-spherical representations with real infinitesimal character, we deduce that the algebraic wavefront set is a singleton, as conjectured by Moeglin and Waldspurger. As a corollary, we obtain an explicit description of the wavefront set of an irreducible spherical representation with real Satake parameter.

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arXiv2022

Some Unipotent Arthur Packets for Reductive $p$-adic Groups

Let $k$ be a $p$-adic field and let $\mathbf{G}(k)$ be the $k$-points of a connected reductive group, inner to split. The set of Aubert-Zelevinsky duals of the constituents of a tempered L-packet form an Arthur packet for $\mathbf{G}(k)$. In this paper, we give an alternative characterization of such Arthur packets in terms of the wavefront set, proving in some instances a conjecture of Jiang-Liu and Shahidi. Pursuing an analogy with real and complex groups, we define some special unions of Arthur packets which we call \emph{weak} Arthur packets and describe their constituents in terms of their Langlands parameters.

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arXiv2022

The Wavefront Sets of Unipotent Supercuspidal Representations

Let $\mathbf{G}(\mathsf{k})$ be a semisimple $p$-adic group, inner to split. In this article, we compute the algebraic and canonical unramified wavefront sets of the irreducible supercuspidal representations of $\mathbf{G}(\mathsf{k})$ in Lusztig's category of unipotent representations.

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arXiv2022

On the generalized Ramanujan conjecture over function fields

Let $G$ be a simple group over a global function field $K$, and let $π$ be a cuspidal automorphic representation of $G$. Suppose $K$ has two places $u$ and $v$ (satisfying a mild restriction on the residue field cardinality), at which the group $G$ is quasi-split, such that $π_u$ is tempered and $π_v$ is unramified and generic. We prove that $π$ is tempered at all unramified places $K_w$ at which $G$ is unramified quasi-split. The proof uses the Galois parametrization of cuspidal representations due to V. Lafforgue to relate the local Satake parameters of $π$ to Deligne's theory of Frobenius weights. The main observation is that, in view of the classification of generic unitary spherical representations, due to Barbasch and the first-named author, the theory of weights excludes generic complementary series as possible local components of $π$. This in turn determines the local Frobenius weights at all unramified places. In order to apply this observation in practice we need a result of the second-named author with Gan and Sawin on the weights of discrete series representations.

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arXiv2021

Deformations of unitary Howe dual pairs

We study deformations of the Howe dual pairs $(\mathrm{U}(n),\mathfrak{u}(1,1))$ and $(\mathrm{U}(n),\mathfrak{u}(2|1))$ to the context of a rational Cherednik algebra $H_{1,c}(G,E)$ associated with a real reflection group $G$ acting on a real vector space $E$ of even dimension. For each pair, we show that the Lie (super)algebra structure of one partner is preserved under the deformation, which leads to a multiplicity-free decomposition of the standard module or its tensor product with a spinor space. For the case where $E$ is two-dimensional and $G$ is a dihedral group, we provide complete descriptions for the deformed pair and the relevant joint-decomposition.

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arXiv2021

Weyl groups, the Dirac inequality, and isolated unitary unramified representations

I present several applications of the Dirac inequality to the determination of isolated unitary representations and associated "spectral gaps" in the case of unramified principal series. The method works particularly well in order to attach irreducible unitary representations to the large nilpotent orbits (e.g., regular, subregular) in the Langlands dual complex Lie algebra. The results could be viewed as a $p$-adic analogue of Salamanca-Riba's classification of irreducible unitary $(\mathfrak g,K)$-modules with strongly regular infinitesimal character.

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