{"title":"Irrationality of the general smooth quartic 3-fold using intermediate Jacobians","authors":"Benson Farb","doi":"10.1016/j.aim.2025.110160","DOIUrl":"10.1016/j.aim.2025.110160","url":null,"abstract":"<div><div>We prove that the intermediate Jacobian of the Klein quartic 3-fold <em>X</em> is not isomorphic, as a principally polarized abelian variety, to a product of Jacobians of curves. As corollaries we deduce (using a criterion of Clemens-Griffiths) that <em>X</em>, as well as the general smooth quartic 3-fold, is irrational. These corollaries were known: Iskovskih-Manin <span><span>[14]</span></span> proved that every smooth quartic 3-fold is irrational. However, the method of proof here is different than that of <span><span>[14]</span></span>, is significantly simpler, and produces an explicit example.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"465 ","pages":"Article 110160"},"PeriodicalIF":1.5,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143421007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Equidimensionality of affine Deligne-Lusztig varieties in mixed characteristic","authors":"Yuta Takaya","doi":"10.1016/j.aim.2025.110153","DOIUrl":"10.1016/j.aim.2025.110153","url":null,"abstract":"<div><div>We prove the equidimensionality of affine Deligne-Lusztig varieties in mixed characteristic. This verifies a conjecture made by Rapoport and implies that the results of Nie and Zhou-Zhu can be extended to the whole irreducible components of affine Deligne-Lusztig varieties. The method is to translate the work of Hartl-Viehmann into mixed characteristic and construct local foliations for affine Deligne-Lusztig varieties. This leads us to develop a theory of formal algebraic geometry for perfect schemes.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"465 ","pages":"Article 110153"},"PeriodicalIF":1.5,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143421021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Quillen cohomology of enriched operads","authors":"Truong Hoang","doi":"10.1016/j.aim.2025.110151","DOIUrl":"10.1016/j.aim.2025.110151","url":null,"abstract":"<div><div>A modern insight due to Quillen, which is further developed by Lurie, asserts that many cohomology theories of interest are particular cases of a single construction, which allows one to define cohomology groups in an abstract setting using only intrinsic properties of the category (or ∞-category) at hand. This universal cohomology theory is known as Quillen cohomology. In any setting, Quillen cohomology of a given object is classified by its cotangent complex. The main purpose of this paper is to study Quillen cohomology of operads enriched over a general base category. Our main result provides an explicit formula for computing Quillen cohomology of enriched operads, based on a procedure of taking certain infinitesimal models of their cotangent complexes. Furthermore, we propose a natural construction of the twisted arrow ∞-categories of simplicial operads. We then assert that the cotangent complex of a simplicial operad can be represented as a spectrum valued functor on its twisted arrow ∞-category.</div><div>When working in stable base categories such as chain complexes and spectra, Francis and Lurie proved the existence of a fiber sequence relating the cotangent complex and Hochschild complex of an <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>-algebra, from which a conjecture of Kontsevich is verified. We establish an analogous fiber sequence for the operad <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> itself, in the topological setting.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"465 ","pages":"Article 110151"},"PeriodicalIF":1.5,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143421006","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The classification of vertex operator algebras of OZ-type generated by Ising vectors of σ-type","authors":"Cuipo Jiang , Ching Hung Lam , Hiroshi Yamauchi","doi":"10.1016/j.aim.2025.110146","DOIUrl":"10.1016/j.aim.2025.110146","url":null,"abstract":"<div><div>We classify vertex operator algebras (VOAs) of OZ-type generated by Ising vectors of <em>σ</em>-type. As a consequence of the classification, we also prove that such VOAs are simple, rational, <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-cofinite and unitary, that is, they have compact real forms generated by Ising vectors of <em>σ</em>-type over the real numbers.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"465 ","pages":"Article 110146"},"PeriodicalIF":1.5,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143421022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Skein and cluster algebras of unpunctured surfaces for sp4","authors":"Tsukasa Ishibashi , Wataru Yuasa","doi":"10.1016/j.aim.2025.110149","DOIUrl":"10.1016/j.aim.2025.110149","url":null,"abstract":"<div><div>As a sequel to our previous work <span><span>[18]</span></span> on the <span><math><msub><mrow><mi>sl</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-case, we introduce a skein algebra <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><msub><mrow><mi>sp</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><mi>Σ</mi></mrow><mrow><mi>q</mi></mrow></msubsup></math></span> consisting of <span><math><msub><mrow><mi>sp</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-webs on a marked surface Σ, incorporating certain “clasped” skein relations at special points. We further investigate its cluster structure. We also define a natural <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-form <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><msub><mrow><mi>sp</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><mi>Σ</mi></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow></msubsup><mo>⊂</mo><msubsup><mrow><mi>S</mi></mrow><mrow><msub><mrow><mi>sp</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><mi>Σ</mi></mrow><mrow><mi>q</mi></mrow></msubsup></math></span>, while the natural coefficient ring <span><math><mi>R</mi></math></span> of <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><msub><mrow><mi>sp</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><mi>Σ</mi></mrow><mrow><mi>q</mi></mrow></msubsup></math></span> includes the inverse of the quantum integer <span><math><msub><mrow><mo>[</mo><mn>2</mn><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span>. We prove that its boundary-localization <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><msub><mrow><mi>sp</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><mi>Σ</mi></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow></msubsup><mo>[</mo><msup><mrow><mo>∂</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>]</mo></math></span> embeds into a quantum cluster algebra <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><msub><mrow><mi>sp</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><mi>Σ</mi></mrow><mrow><mi>q</mi></mrow></msubsup></math></span> that quantizes the function ring of the moduli space <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>S</mi><msub><mrow><mi>p</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><mi>Σ</mi></mrow><mrow><mo>×</mo></mrow></msubsup></math></span>. Furthermore, we establish the positivity of Laurent expressions of elevation-preserving webs, following an approach similar to <span><span>[18]</span></span>. We also propose a characterization of cluster variables in the spirit of Fomin–Pylyavskyy <span><span>[9]</span></span> using <span><math><msub><mrow><mi>sp</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-webs, and provide infinitely many supporting examples on a quadrilateral.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"465 ","pages":"Article 110149"},"PeriodicalIF":1.5,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143403349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Derived delooping levels and finitistic dimension","authors":"Ruoyu Guo, Kiyoshi Igusa","doi":"10.1016/j.aim.2025.110152","DOIUrl":"10.1016/j.aim.2025.110152","url":null,"abstract":"<div><div>In this paper, we develop new ideas regarding the finitistic dimension conjecture, or the findim conjecture for short. Specifically, we improve upon the delooping level by introducing three new invariants called the effective delooping level edell, the sub-derived delooping level <span><math><mrow><mi>sub</mi></mrow><mtext>-</mtext><mrow><mi>ddell</mi></mrow></math></span>, and the derived delooping level ddell. They are all better upper bounds for the opposite Findim. Precisely, we prove<span><span><span><math><mrow><mi>Findim</mi></mrow><mspace></mspace><msup><mrow><mi>Λ</mi></mrow><mrow><mi>op</mi></mrow></msup><mo>=</mo><mrow><mi>edell</mi></mrow><mspace></mspace><mi>Λ</mi><mo>≤</mo><mrow><mi>ddell</mi></mrow><mspace></mspace><mi>Λ</mi><mspace></mspace><mo>(</mo><mtext>or </mtext><mrow><mi>sub</mi></mrow><mtext>-</mtext><mrow><mi>ddell</mi></mrow><mspace></mspace><mi>Λ</mi><mo>)</mo><mo>≤</mo><mrow><mi>dell</mi></mrow><mspace></mspace><mi>Λ</mi></math></span></span></span> and provide examples where the last inequality is strict (including the recent example from <span><span>[16]</span></span> where <span><math><mrow><mi>dell</mi></mrow><mspace></mspace><mi>Λ</mi><mo>=</mo><mo>∞</mo></math></span>, but <span><math><mrow><mi>ddell</mi></mrow><mspace></mspace><mi>Λ</mi><mo>=</mo><mn>1</mn><mo>=</mo><mrow><mi>Findim</mi></mrow><mspace></mspace><msup><mrow><mi>Λ</mi></mrow><mrow><mi>op</mi></mrow></msup></math></span>).</div><div>We further enhance the connection between the findim conjecture and tilting theory by showing finitely generated modules with finite derived delooping level form a torsion-free class <span><math><mi>F</mi></math></span>. Therefore, studying the corresponding torsion pair <span><math><mo>(</mo><mi>T</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> will shed more light on the little finitistic dimension. Lastly, we relate the delooping level to the <em>ϕ</em>-dimension <em>ϕ</em>dim, a popular upper bound for findim, and recover a sufficient condition for the findim conjecture given in <span><span>[5]</span></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"464 ","pages":"Article 110152"},"PeriodicalIF":1.5,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143394543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"SYZ for index 1 hypersurfaces in projective space","authors":"Mohamed El Alami","doi":"10.1016/j.aim.2025.110144","DOIUrl":"10.1016/j.aim.2025.110144","url":null,"abstract":"<div><div>We study homological mirror symmetry of the singular hypersurface <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mi>V</mi><mo>(</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>−</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>⊆</mo><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span>. Following an SYZ type approach, we produce an LG-model whose Fukaya-Seidel category recovers line bundles on <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. As a byproduct of our approach, we answer a conjecture of N. Sheridan about generating the small component of the Fukaya category of the <em>smooth</em> index 1 Fano hypersurface in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> without bounding co-chains.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"464 ","pages":"Article 110144"},"PeriodicalIF":1.5,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143388286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Constructions of Turán systems that are tight up to a multiplicative constant","authors":"Oleg Pikhurko","doi":"10.1016/j.aim.2025.110148","DOIUrl":"10.1016/j.aim.2025.110148","url":null,"abstract":"<div><div>For positive integers <span><math><mi>n</mi><mo>⩾</mo><mi>s</mi><mo>></mo><mi>r</mi></math></span>, the <em>Turán function</em> <span><math><mi>T</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span> is the smallest size of an <em>r</em>-graph with <em>n</em> vertices such that every set of <em>s</em> vertices contains at least one edge. Also, define the <em>Turán density</em> <span><math><mi>t</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span> as the limit of <span><math><mi>T</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo>)</mo><mo>/</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>r</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span> as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>. The question of estimating these parameters received a lot of attention after it was first raised by Turán in 1941. A trivial lower bound is <span><math><mi>t</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo>)</mo><mo>⩾</mo><mn>1</mn><mo>/</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>s</mi></mtd></mtr><mtr><mtd><mrow><mi>s</mi><mo>−</mo><mi>r</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span>. In the 1990s, de Caen conjectured that <span><math><mi>r</mi><mo>⋅</mo><mi>t</mi><mo>(</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>r</mi><mo>)</mo><mo>→</mo><mo>∞</mo></math></span> as <span><math><mi>r</mi><mo>→</mo><mo>∞</mo></math></span> and offered 500 Canadian dollars for resolving this question.</div><div>We disprove this conjecture by showing more strongly that for every integer <span><math><mi>R</mi><mo>⩾</mo><mn>1</mn></math></span> there is <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span> (in fact, <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span> can be taken to grow as <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mspace></mspace><mi>R</mi><mi>ln</mi><mo></mo><mi>R</mi></math></span>) such that <span><math><mi>t</mi><mo>(</mo><mi>r</mi><mo>+</mo><mi>R</mi><mo>,</mo><mi>r</mi><mo>)</mo><mo>⩽</mo><mo>(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mo>/</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>r</mi><mo>+</mo><mi>R</mi></mrow></mtd></mtr><mtr><mtd><mi>R</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span> as <span><math><mi>r</mi><mo>→</mo><mo>∞</mo></math></span>, that is, the trivial lower bound is tight for every <em>R</em> up to a multiplicative constant <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"464 ","pages":"Article 110148"},"PeriodicalIF":1.5,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143388288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The affine Springer fiber – sheaf correspondence","authors":"Eugene Gorsky , Oscar Kivinen , Alexei Oblomkov","doi":"10.1016/j.aim.2025.110143","DOIUrl":"10.1016/j.aim.2025.110143","url":null,"abstract":"<div><div>Given a semisimple element in the loop Lie algebra of a reductive group, we construct a quasi-coherent sheaf on a partial resolution of the trigonometric commuting variety of the Langlands dual group. The construction uses affine Springer theory and can be thought of as an incarnation of 3d mirror symmetry. For the group <span><math><mi>G</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, the corresponding partial resolution is <span><math><msup><mrow><mi>Hilb</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>×</mo></mrow></msup><mo>×</mo><mi>C</mi><mo>)</mo></math></span>. We also consider a quantization of this construction for homogeneous elements.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"464 ","pages":"Article 110143"},"PeriodicalIF":1.5,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Proof of a conjecture of Kudla and Rallis on quotients of degenerate principal series","authors":"Johannes Droschl","doi":"10.1016/j.aim.2025.110145","DOIUrl":"10.1016/j.aim.2025.110145","url":null,"abstract":"<div><div>In this paper we prove a conjecture of Kudla and Rallis, see <span><span>[12, Conjecture V.3.2]</span></span>. Let <em>χ</em> be a unitary character, <span><math><mi>s</mi><mo>∈</mo><mi>C</mi></math></span> and <em>W</em> a symplectic vector space over a non-archimedean field with symmetry group <span><math><mi>G</mi><mo>(</mo><mi>W</mi><mo>)</mo></math></span>. Denote by <span><math><mi>I</mi><mo>(</mo><mi>χ</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> the degenerate principal series representation of <span><math><mi>G</mi><mo>(</mo><mi>W</mi><mo>⊕</mo><mi>W</mi><mo>)</mo></math></span>. Pulling back <span><math><mi>I</mi><mo>(</mo><mi>χ</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> along the natural embedding <span><math><mi>G</mi><mo>(</mo><mi>W</mi><mo>)</mo><mo>×</mo><mi>G</mi><mo>(</mo><mi>W</mi><mo>)</mo><mo>↪</mo><mi>G</mi><mo>(</mo><mi>W</mi><mo>⊕</mo><mi>W</mi><mo>)</mo></math></span> gives a representation <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>W</mi><mo>,</mo><mi>W</mi></mrow></msub><mo>(</mo><mi>χ</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> of <span><math><mi>G</mi><mo>(</mo><mi>W</mi><mo>)</mo><mo>×</mo><mi>G</mi><mo>(</mo><mi>W</mi><mo>)</mo></math></span>. Let <em>π</em> be an irreducible smooth complex representation of <span><math><mi>G</mi><mo>(</mo><mi>W</mi><mo>)</mo></math></span>. We then prove<span><span><span><math><msub><mrow><mi>dim</mi></mrow><mrow><mi>C</mi></mrow></msub><mo></mo><msub><mrow><mi>Hom</mi></mrow><mrow><mi>G</mi><mo>(</mo><mi>W</mi><mo>)</mo><mo>×</mo><mi>G</mi><mo>(</mo><mi>W</mi><mo>)</mo></mrow></msub><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>W</mi><mo>,</mo><mi>W</mi></mrow></msub><mo>(</mo><mi>χ</mi><mo>,</mo><mi>s</mi><mo>)</mo><mo>,</mo><mi>π</mi><mo>⊗</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>∨</mo></mrow></msup><mo>)</mo><mo>=</mo><mn>1</mn><mo>.</mo></math></span></span></span> We also give analogous statements for <em>W</em> orthogonal or unitary. This gives in particular a new proof of the conservation relation of the local theta correspondence for symplectic-orthogonal and unitary dual pairs.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"464 ","pages":"Article 110145"},"PeriodicalIF":1.5,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143277974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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