Pith. sign in

REVIEW 4 major objections 4 minor 32 references

The spectrum of the symplectic Grassmannian and $\mathrm{Mat}_{n,m}$

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper computes the full multiplicity-free spectrum of the matrix space and the symplectic Grassmannian, and derives Howe duality in type II, explicit local Miyawaki lifts, and a pole formula for Godement–Jacquet L-functions.

desk verdict A genuinely new derivative-plus-Local-Structure method with real payoffs, but the load-bearing Corollary 3.1 and the metaplectic transfer need full proofs before the results can be trusted. read the letter →

arxiv 2506.13530 v2 pith:S4ZZV37H submitted 2025-06-16 math.RT

Johannes Droschl This is my paper · ORCID
classification math.RT MSC 22E5014M2720G25
keywords sphericalvarietiesrho-derivativesmultiplicity-freespectrumsymplecticGrassmannianspaceofmatricesHowedualityMiyawakiliftingsGodement–JacquetL-functions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper gives a unified way to compute the full spectrum—the list of irreducible representations that can be realized on Schwartz functions—of two spherical varieties over a local non-archimedean field: the space $\mathrm{Mat}_{n,m}$ of $n\times m$ matrices under $\mathrm{GL}_n\times\mathrm{GL}_m$, and the symplectic Grassmannian $\mathrm{SGr}_{n,m}$ under $\mathrm{Sp}_n\times\mathrm{Sp}_m$. The main theorems assert that both spaces are multiplicity-free, have the lifting property, and that their spectra have explicit descriptions in terms of the socle of a single induced representation built from a rank datum attached to $\pi$. From these descriptions the paper derives a new proof of Howe duality in type II, an explicit formula for local Miyawaki lifts in the Hilbert–Siegel case, and a pole formula for Godement–Jacquet $L$-functions at the point $s=-(n-1)/2$. The method combines the theory of $\rho$-derivatives with the Local Structure Theorem for spherical varieties, and the same arguments are stated to extend to metaplectic covers and inner forms.

What carries the argument

The load-bearing mechanism is an induction on rank strata rooted in the theory of $\rho$-derivatives and the Local Structure Theorem for spherical varieties. For each orbit stratum $\mathrm{Mat}^r_{n,m}$ or $\mathrm{SGr}^r_{n,m}$, the theorem gives an equivariant isomorphism $N_Y\times S_Y\to X^\circ_Y$ with $S_Y\cong\mathrm{Mat}_{n-r,m-r}\times\mathrm{GL}_r$ or $S_Y\cong\mathrm{Mat}_{n-r,m-r}\times\mathrm{Sp}_r$, which transfers the spectrum problem from the whole variety to its boundary strata; the lifting property is then proven by Frobenius reciprocity and a genericity argument. The single most important auxiliary result is Corollary 3.1: for a self-dual segment $\Delta$, $\mathrm{soc}(Z(\Delta)\rtimes\pi)$ is multiplicity-free of length at most two, and any embedding of $\tau$ into both $Z(\Delta)\rtimes\pi$ and $Z(\Delta)\rtimes\pi'$ forces $\pi\cong\pi'$. This makes the map $T^m_n(\pi,\mu)$ well-defined, and the paper states that the same arguments carry over to $\mathrm{Mp}_n$ by adjusting notation.

What would settle it

Find non-isomorphic $\pi,\pi'\in\mathrm{Irr}(\mathrm{Sp}_n)$ and a self-dual segment $\Delta$ such that $\mathrm{soc}(Z(\Delta)\rtimes\pi)$ and $\mathrm{soc}(Z(\Delta)\rtimes\pi')$ share an irreducible constituent. Corollary 3.1 forbids this, and since the definition of $T^m_n(\pi,\mu)$ uses exactly that uniqueness, such an example would collapse the symplectic spectrum formula.

Watch

Extended reading notes

Core claim

The central discovery is that the spectrum of $\mathrm{Mat}_{n,m}$ is exactly $\{\pi\otimes T^m_n(\pi):\pi\in\mathrm{Irr}(\mathrm{GL}_n)\}$, where for the minimal rank $r$ such that $\pi$ embeds into $\nu^{r/2}_{n-r}\times\tau$, one sets $T^m_n(\pi)=\mathrm{soc}(\tau^\vee\nu^{(m-n)/2}\times\nu^{-r/2}_{m-r})$; similarly, the spectrum of $(\mathrm{SGr}_{n,m},L_\mu)$ is exactly $\{\pi\otimes\pi':\pi\otimes\pi'\subseteq\pi\otimes T^m_n(\pi,\mu)\}$, where $T^m_n(\pi,\mu)=\mathrm{soc}(\mu(\det_{m-n+r})^{-1}\nu^{-r/2}_{m-n+r}\rtimes\tau^\vee)$ for the maximal $r$ with $\pi\hookrightarrow\mu(\det_r)^{-1}\nu^{-(m-n+r)/2}_r\rtimes\tau$. The paper proves both spaces are multiplicity-free and have the lifting property, and shows that this implies Howe duality in type II, that the local Miyawaki lift $M_\mu^m(\pi)$ has cosocle $\mathrm{soc}(\mu_\psi(\widetilde{\det}_{m-n+r})\nu^{r/2}_{m-n+r}\rtimes\tau)$ of length at most two, and that $\mathrm{ord}_{s=-(n-1)/2}L(\pi,s)=\Lambda(\nu^{r/2}_{n-r},\tau)+1$.

Load-bearing premise

The argument depends on Corollary 3.1, that for a self-dual segment $\Delta$ and any irreducible $\pi$ of $\mathrm{Sp}_n$ the socle of $Z(\Delta)\rtimes\pi$ is multiplicity-free of length at most two and determines $\pi$ uniquely; the proof is a compressed induction, and its stated analog for the metaplectic cover $\mathrm{Mp}_n$ is asserted without a written proof.

Editorial extensions

If this is right

  • Every irreducible smooth representation of $\mathrm{GL}_n$ appears exactly once in the spectrum of $\mathrm{Mat}_{n,m}$, paired with the explicit partner $T^m_n(\pi)$, so the matrix space is multiplicity-free with a rank-by-rank construction.
  • Every $\pi\in\mathrm{Irr}(\mathrm{Sp}_n)$ has at most two possible partners in the spectrum of $(\mathrm{SGr}_{n,m},L_\mu)$, both contained in $\pi\otimes T^m_n(\pi,\mu)$, and each orbit stratum contributes a definite tensor product of socles.
  • Howe duality in type II follows as a corollary: the big theta lift $\Theta_{n,m}(\pi)$ has irreducible cosocle $\theta_{n,m}(\pi)$, and $\theta_{n,m}(\pi)\cong\theta_{n,m}(\pi')\neq0$ implies $\pi\cong\pi'$.
  • Local Miyawaki lifts in the Hilbert–Siegel case become explicit: $M_\mu^m(\pi)$ is of finite length, nonzero for $m\geq n$, and its cosocle is the socle of a single explicitly written induced representation of length at most two; for $n\geq m$ the correspondence is injective.
  • The pole of the Godement–Jacquet $L$-function at $s=-(n-1)/2$ is computed geometrically: its order equals $\Lambda(\nu^{r/2}_{n-r},\tau)+1$, where $r$ is the first rank stratum on which the canonical map $T_\pi$ does not vanish.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the same two-step scheme—ordered orbit stratification plus derivative-based genericity—applies to other spherical varieties whose Local Structure Theorem slices are products of matrix spaces and smaller copies of the same group, the spectra of those varieties would be explicitly computable by the same induction.
  • The explicit pole formula suggests a testable pattern beyond $\mathrm{Mat}_{n,n}$: for more general spherical varieties, the order of a pole of a relative $L$-function may be read off from the first nonvanishing orbit stratum of the canonical period map.
  • The extension to metaplectic groups is asserted by analogy rather than written out in full; since Corollary 3.1 is the hinge, a careful written verification for $\mathrm{Mp}_n$ would be the natural next check.
  • The Miyawaki description may allow one to compute the full big lift $M_\mu^m(\pi)$ rather than only its cosocle, because the proof obtains the cosocle from the image of an intertwining operator whose poles are already controlled.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes a general strategy for computing the spectrum of spherical varieties over non-archimedean local fields, combining rho-derivatives, the Local Structure Theorem, and Frobenius reciprocity. It applies this strategy to the space of matrices Mat_{n,m} and to the symplectic Grassmannian SGr_{n,m}, claiming multiplicity-freeness, the lifting property, and explicit descriptions of their spectra: for Mat_{n,m} the spectrum is {pi tensor T^m_n(pi)} and for SGr_{n,m} it is described in terms of pi tensor T^m_n(pi,mu). From these results the paper derives a new proof of Howe duality in type II, an explicit description of local Miyawaki liftings in the Hilbert-Siegel case, and a formula for the order of poles of Godement-Jacquet L-functions at s=-(n-1)/2. It further claims that the arguments extend to metaplectic covers of GL_n and Sp_n and to inner forms of GL_n.

Significance. If the main theorems are correct, this is a substantial contribution: it gives a unified, derivative-theoretic derivation of two nontrivial spectra, a new proof of Howe duality in type II, explicit local Miyawaki lifts, and a new route to the Godement-Jacquet L-factor pole formula that does not use the functional equation. The paper also proposes a general framework that could apply to other spherical varieties. The claimed extensions to metaplectic covers and division algebras are potentially significant for the local theta correspondence. However, several load-bearing steps are compressed or asserted by analogy, in particular the symplectic Delta-derivative theorem (Corollary 3.1), the transfer to metaplectic groups, and the equality statement in the symplectic Grassmannian spectrum (Proposition 5.1). The current manuscript therefore does not yet fully substantiate its main claims; it is a promising but incomplete draft.

major comments (4)
  1. [Section 3.3, Corollary 3.1] The definition of T^m_n(pi,mu) immediately before Section 3.4 relies on Corollary 3.1 for well-definedness, and Theorem 5.2 and Proposition 5.1 rely on it for the spectrum of SGr_{n,m}. The proof of Corollary 3.1 for non-self-dual segments is an induction that invokes the identity soc(Z(Delta)⋊pi) = soc((rho nu^a_rho)^{1+d_{rho nu^a_rho,max}(pi)} ⋊ (soc(Z(-Delta)⋊ D^max_{rho nu^a_rho}(pi)))) with the justification "we saw in the proof of Lemma 3.12". However, Lemma 3.12 is not proved in the text (its proof is said to be analogous to Lemma 3.8, a GL_n statement), and the displayed formula does not by itself demonstrate that the resulting socle remains multiplicity-free and that an embedding into both Z(Delta)⋊pi and Z(Delta)⋊pi' forces pi ≅ pi'. Please provide a complete proof of Corollary 3.1, or precise references for every step of the induction.
  2. [Sections 3.2.2, 3.3, 5.3.3] The transfer of all Sp_n results to the metaplectic group Mp_n is asserted in one sentence ("it is just a matter of adjusting the notation"), and similar blanket assertions are made for GL'_n and for the metaplectic GL_n. This transfer is load-bearing for Theorems 5.3-5.5 and especially for Theorem 5.7 on local Miyawaki liftings, where the condition on the mu_2-action appears explicitly. On metaplectic covers, genuine representations and Weil indices can affect Jacquet modules and socle decompositions; the paper should either prove the needed analogues or give a precise statement of which results carry over and why. As written, this is an unverified assumption rather than a proved extension.
  3. [Section 5.3.2, Lemma 5.6 and Proposition 5.1] Lemma 5.6, the symplectic analogue of Lemma 5.3, is the key input for Theorem 5.2, but its proof is summarized as "We proceed analogously to Lemma 5.3". The symplectic case has the additional feature of self-dual segments and length-two socles, so the analogy is not immediate and needs to be spelled out. Moreover, Proposition 5.1 upgrades the inclusion in Theorem 5.2 to an equality by a limiting argument in s; the proof asserts that the order of the pole of f_s at s=0 equals the order of the associated intertwining operator, and then identifies cosoc(Im(f_0)) with soc(Pi'_0) using Theorem 3.3. As written, these steps are too compressed for the main spectrum theorem; please expand the argument.
  4. [Section 5.4.3, Lemma 5.11 and Theorem 5.8] The proof of Lemma 5.11 claims that the pole order of L(pi,s) at s=-(nd-1)/2 equals that of L(rho_{n,alpha},s), based on a factorization of the Godement-Jacquet zeta integral. The displayed identity involves an integral over Mat_{n-|alpha|,|alpha|} × Mat_{|alpha|,n-|alpha|} × GL_{|alpha|}; the argument requires that this integral can be chosen non-zero and that no other contributions affect the pole. This is not fully justified, and Theorem 5.8 depends on it. Please provide the missing quantitative statement.
minor comments (4)
  1. [Section 2.1 and Sections 3.2.1/3.2.2] The displayed definitions of pi^{×k} and of the iterated derivatives D^k_{rho,r}(pi) and D^k_rho(pi) contain corrupted symbols (e.g., "⌟⟪⟨⟪rl⟫l⟩⟩"); these should be replaced by standard notation.
  2. [Introduction and Section 5.3.1] There are typos such as "enabeling" in the introduction and missing spaces in the sentence "Itisclearthatthecompositionisgivenbyintegrating..." in Lemma 5.5; the latter should also be rewritten for clarity.
  3. [Section 5.3.1, Lemma 5.5] The chain of equalities in the proof of Lemma 5.5 appears to contain a typo: it should read T(chi_{k-1}) = T(chi_k) + T(chi'), not T(chi_k) = T(chi_{k-1}) = T(chi') + T(chi_k).
  4. [Section 5.3.3] The notation "Irr̃SGrm,n,Lµ" in Theorem 5.4 is undefined and should be written consistently with the notation used elsewhere, e.g., Irr_{SGr_{n,m},L_mu}(Mp_n × Mp_m).

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the spectrum theorems are derived from external derivative theory and the Local Structure Theorem, with only minor non-load-bearing self-citations and some asserted technical transfers.

full rationale

I walked the derivation chain. Theorem 5.1 and Theorem 5.2 are obtained from external derivative theory (Lemmas 2 and 3, plus [AM23], [LM18], [LM25]), the Geometric Lemma, the Local Structure Theorem, and the paper's own Lemmas 5.3–5.5 and 5.6–5.8; none of these results is equivalent to the spectra being proved. The operators T_n^m and T_n^m(pi,mu) are defined after, and independently of, the spectral statements, and their well-definedness rests on Corollary 3.1, which is proved in the text rather than assumed. The self-citations to [Dro25b] and [Dro25a] are contextual or supplementary, for instance 'see also [Dro25b, Lemma 2.9.3]' in Lemma 3.13, whose proof is self-contained; they are not load-bearing for the new proof of type II Howe duality. The main caveats are rigor gaps rather than circularity: Corollary 3.1's proof invokes a socle identity 'we saw in the proof of Lemma 3.12' although Lemma 3.12 is only stated as an analogue of Lemma 3.8, and the transfer to the metaplectic group is asserted by 'all results here have analogous statements when one replaces the group Sp_n by Mp_n.' These passages are in-scope omissions or asserted transfers, but they do not make the output equal to an input. The pole formula in Theorem 5.8 is derived from zeta-integral factorization and intertwining-operator pole computations, not by defining the L-function as that pole. I therefore find no circular step; the score of 2 records only minor non-load-bearing self-citations and the noted non-circular technical gaps.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The derivations rest on standard classifications, derivative theory, the Local Structure Theorem, and the Geometric Lemma; no empirical constants are fitted. The only non-standard input is the unproved assertion that the metaplectic and division algebra cases follow by the same arguments.

assumptions (8)
  • standard math Zelevinsky classification of smooth irreducible representations of GL_n and its extension to inner forms (Theorem 2.4, [Zel80], [LM16]).
    Used throughout to identify socles and derivatives of representations.
  • domain assumption Atobe-Minguez theory of rho-derivatives for classical groups (Lemmas 3.5 through 3.12, [AM23]).
    The main technical engine of the paper; its validity is assumed and cited.
  • standard math Local Structure Theorem for spherical varieties (Theorem 4.2, [LV83], [BLV86], [KK24]).
    Provides the local form S_Y used in the inductive argument.
  • standard math Luna-Vust classification of spherical embeddings by colored fans (Theorem 4.1, [LV83], [Per14]).
    Used to prove Lemma 5.1 identifying the ordered spherical variety as Mat^{<=r}.
  • standard math Geometric Lemma for Jacquet modules of parabolically induced representations ([BZ77], [Tad95]).
    Used in Hom-space computations in Lemmas 5.3 and 5.6 and in intertwining operator proofs.
  • domain assumption For symplectic and metaplectic groups, char F is not 2 (Section 2.3).
    Needed for the metaplectic cover and theta-correspondence setup.
  • domain assumption All G-orbits are defined over F, assumed for simplicity (Section 4).
    The paper notes the assumption and states that all examples satisfy it.
  • ad hoc to paper The results extend unchanged to metaplectic covers and division algebras (Theorems 5.3 through 5.5).
    These theorems are stated without proof, relying on an unverified assertion that every step carries over.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The spectrum of the symplectic Grassmannian and $\mathrm{Mat}_{n,m}$." pith.science (2026). https://pith.science/paper/S4ZZV37H

@misc{pith2026250613530,
  author       = {Pith},
  title        = {Pith review of: The spectrum of the symplectic Grassmannian and $\mathrmMat_n,m$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S4ZZV37H}},
  note         = {Machine review of arXiv:2506.13530}
}
abstract

Let $\mathbf{G}$ be a reductive group and $\mathbf{X}$ a spherical $\mathbf{G}$-variety over a local non-archimedean field $\mathbb{F}$. We denote by $S(\mathbf{X}(\mathbb{F}))$ the Schwartz-functions on $\mathbf{X}(\mathbb{F})$. In this paper we offer a new approach on how to obtain bounds on \[\dim_{\mathbb{C}}\mathrm{Hom}_{\mathbf{G}(\mathbb{F})}(S(\mathbf{X}(\mathbb{F})),\pi)\]for an irreducible smooth representation $\pi$ of $\mathbf{G}(\mathbb{F})$. Our strategy builds on the theory of $\rho$-derivatives and the Local Structure Theorem for spherical varieties. Currently, we focus on the case of the symplectic Grassmannian and the space of matrices. In particular, we obtain a new proof of Howe duality in type II as well as an explicit description of the local Miyawaki-liftings in the Hilbert-Siegel case. Furthermore, we manage to extend previous results of the author regarding the conservation relation in the theta correspondence to metaplectic covers of symplectic groups. Finally, we use our new proof of Howe duality in type II to relate the order of the poles of Godement-Jacquet $L$-functions to the geometry of the space of matrices and the order of poles of certain intertwining operators.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

32 extracted references · 31 canonical work pages

  1. [1]

    Hiraku Atobe, Wee Teck Gan, Atsushi Ichino, Tasho Kaletha, Alberto Mínguez, and Sug Woo Shin, Local intertwining relations and co-tempered a -packets of classical groups, 2024, arXiv:2410.13504

    arXiv 2024

  2. [2]

    Hiraku Atobe and Alberto M\'inguez, The explicit Z elevinsky- A ubert duality , Compos. Math. 159 (2023), no. 2, 380--418. 4549708

    work page 2023

  3. [3]

    Hiraku Atobe, A theory of M iyawaki liftings: the H ilbert– S iegel case , Mathematische Annalen 376 (2019), 1467 -- 1535

    work page 2019

  4. [4]

    Brion, D

    M. Brion, D. Luna, and Th. Vust, Espaces homog\`enes sph\'eriques, Invent. Math. 84 (1986), no. 3, 617--632. 837530

    work page 1986

  5. [5]

    55 (1986), no

    Michel Brion, Quelques propri\'et\'es des espaces homog\`enes sph\'eriques, Manuscripta Math. 55 (1986), no. 2, 191--198. 833243

    work page 1986

  6. [6]

    I. N. Bernstein and A. V. Zelevinsky, Representations of the group GL(n,F), where F is a local non- A rchimedean field , Uspehi Mat. Nauk 31 (1976), no. 3(189), 5--70. 425030

    work page 1976

  7. [7]

    , Induced representations of reductive p -adic groups. I , Ann. Sci. \' E cole Norm. Sup. (4) 10 (1977), no. 4, 441--472. 579172

    work page 1977

  8. [8]

    David Ben-Zvi, Yiannis Sakellaridis, and Akshay Venkatesh, Relative langlands duality, 2024

    work page 2024

Show all 32 references
  1. [9]

    Johannes Droschl, A note on the howe duality conjecture for symplectic-orthogonal and unitary pairs, 2025, arXiv:2503.08475

    2025

  2. [10]

    , Proof of a conjecture of K udla and R allis on quotients of degenerate principal series , Advances in Mathematics 464 (2025), 110145

    2025

  3. [11]

    1254, Springer-Verlag, Berlin, 1987

    Stephen Gelbart, Ilya Piatetski-Shapiro, and Stephen Rallis, Explicit constructions of automorphic L -functions , Lecture Notes in Mathematics, vol. 1254, Springer-Verlag, Berlin, 1987. 892097

    1987

  4. [12]

    II , Represent

    Chris Jantzen, On square-integrable representations of classical p -adic groups. II , Represent. Theory 4 (2000), 127--180. 1789464

    2000

  5. [13]

    Kaplan, E

    E. Kaplan, E. Lapid, and J. Zou, Classification of irreducible representations of metaplectic covers of the general linear group over a non-archimedean local field, arXiv preprint arXiv:2206.14731 (2022), N/A

    2022 arXiv

  6. [14]

    Friedrich Knop, Weylgruppe und M omentabbildung , Invent. Math. 99 (1990), no. 1, 1--23. 1029388

    1990

  7. [15]

    , Automorphisms, root systems, and compactifications of homogeneous varieties, J. Amer. Math. Soc. 9 (1996), no. 1, 153--174. 1311823

    1996

  8. [16]

    Kudla and S

    S. Kudla and S. Rallis, On first occurrence in the local theta correspondence, Automorphic representations, L -functions and applications: progress and prospects, Ohio State Univ. Math. Res. Inst. Publ., vol. 11, de Gruyter, Berlin, 2005, pp. 273--308. 2192827

    2005

  9. [17]

    Kudla, On the local theta-correspondence, Invent

    S. Kudla, On the local theta-correspondence, Invent. Math. 83 (1986), no. 2, 229--255. 818351

    1986

  10. [18]

    Lapid and A

    E. Lapid and A. M\' nguez, Geometric conditions for -irreducibility of certain representations of the general linear group over a non-archimedean local field , Adv. Math. 339 (2018), 113--190. 3866895

    2018

  11. [19]

    Erez Lapid and Alberto M\'inguez, A binary operation on irreducible components of L usztig's nilpotent varieties II : applications and conjectures for representations of GL _n over a non-archimedean local field , Pure Appl. Math. Q. 21 (2025), no. 2, 813--863. 4847251

    2025

  12. [20]

    Erez M. Lapid and Stephen Rallis, On the local factors of representations of classical groups, Automorphic representations, L -functions and applications: progress and prospects, Ohio State Univ. Math. Res. Inst. Publ., vol. 11, de Gruyter, Berlin, 2005, pp. 309--359. 2192828

    2005

  13. [21]

    Luna and Th

    D. Luna and Th. Vust, Plongements d'espaces homog\`enes, Comment. Math. Helv. 58 (1983), no. 2, 186--245. 705534

    1983

  14. [22]

    M\' nguez, Correspondance de H owe explicite: paires duales de type II , Ann

    A. M\' nguez, Correspondance de H owe explicite: paires duales de type II , Ann. Sci. \' E c. Norm. Sup\' e r. (4) 41 (2008), no. 5, 717--741. 2504432

    2008

  15. [23]

    Reine Angew

    Alberto M\' nguez, Sur l'irr\' e ductibilit\' e d'une induite parabolique , J. Reine Angew. Math. 629 (2009), 107--131. 2527415

    2009

  16. [24]

    M glin, M.-F

    C. M glin, M.-F. Vign\' e ras, and J.-L. Waldspurger, Correspondances de H owe sur un corps p -adique , Lecture Notes in Mathematics, vol. 1291, Springer-Verlag, Berlin, 1987. 1041060

    1987

  17. [25]

    Groups 19 (2014), no

    Nicolas Perrin, On the geometry of spherical varieties, Transform. Groups 19 (2014), no. 1, 171--223. 3177371

    2014

  18. [26]

    Yiannis Sakellaridis, On the unramified spectrum of spherical varieties over p -adic fields , Compos. Math. 144 (2008), no. 4, 978--1016. 2441254

    2008

  19. [27]

    , Spherical varieties, functoriality, and quantization, I CM --- I nternational C ongress of M athematicians. V ol. 4. S ections 5--8, EMS Press, Berlin, [2023] 2023, pp. 2998--3037. 4680350

    2023

  20. [28]

    396, viii+360

    Yiannis Sakellaridis and Akshay Venkatesh, Periods and harmonic analysis on spherical varieties, Ast\'erisque (2017), no. 396, viii+360. 3764130

    2017

  21. [29]

    Tadi\' c , Structure arising from induction and J acquet modules of representations of classical p -adic groups , J

    M. Tadi\' c , Structure arising from induction and J acquet modules of representations of classical p -adic groups , J. Algebra 177 (1995), no. 1, 1--33. 1356358

    1995

  22. [30]

    Waldspurger, D\' e monstration d'une conjecture de dualit\' e de H owe dans le cas p -adique, p 2 , Festschrift in honor of I

    J.-L. Waldspurger, D\' e monstration d'une conjecture de dualit\' e de H owe dans le cas p -adique, p 2 , Festschrift in honor of I . I . P iatetski- S hapiro on the occasion of his sixtieth birthday, P art I ( R amat A viv, 1989), Israel Math. Conf. Proc., vol. 2, Weizmann, J...

    1989

  23. [31]

    , La formule de P lancherel pour les groupes p -adiques (d'apr\`es H arish- C handra) , J. Inst. Math. Jussieu 2 (2003), no. 2, 235--333. 1989693

    2003

  24. [32]

    A. V. Zelevinsky, Induced representations of reductive p -adic groups. II . O n irreducible representations of GL (n) , Ann. Sci. \' E cole Norm. Sup. (4) 13 (1980), no. 2, 165--210. 584084

    1980

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.