[{"article_type":"original","year":"2021","volume":149,"keyword":["Applied Mathematics","General Mathematics"],"article_processing_charge":"No","publication_identifier":{"issn":["0002-9939"],"eissn":["1088-6826"]},"external_id":{"arxiv":["1910.08286"]},"author":[{"id":"70B7FDF6-608D-11E9-9333-8535E6697425","last_name":"Brown","first_name":"Adam","full_name":"Brown, Adam"},{"last_name":"Romanov","first_name":"Anna","full_name":"Romanov, Anna"}],"date_updated":"2021-02-04T11:25:17Z","page":"37-52","publication_status":"published","day":"01","title":"Contravariant forms on Whittaker modules","month":"01","abstract":[{"text":"Let g be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker g-modules Y(χ,η) introduced by Kostant. We prove that the set of all contravariant forms on Y(χ,η) forms a vector space whose dimension is given by the cardinality of the Weyl group of g. We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules M(χ,η) introduced by McDowell.","lang":"eng"}],"_id":"8773","project":[{"grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"ISTplus - Postdoctoral Fellowships"}],"status":"public","intvolume":" 149","oa":1,"oa_version":"Preprint","doi":"10.1090/proc/15205","date_published":"2021-01-01T00:00:00Z","main_file_link":[{"url":"https://arxiv.org/abs/1910.08286","open_access":"1"}],"acknowledgement":"We would like to thank Peter Trapa for useful discussions, and Dragan Milicic and Arun Ram for valuable feedback on the structure of the paper. The first author acknowledges the support of the European Unions Horizon 2020 research and innovation programme under the Marie Skodowska-Curie Grant Agreement No. 754411. The second author is\r\nsupported by the National Science Foundation Award No. 1803059.","publisher":"American Mathematical Society","citation":{"chicago":"Brown, Adam, and Anna Romanov. “Contravariant Forms on Whittaker Modules.” *Proceedings of the American Mathematical Society*. American Mathematical Society, 2021. https://doi.org/10.1090/proc/15205.","short":"A. Brown, A. Romanov, Proceedings of the American Mathematical Society 149 (2021) 37–52.","ieee":"A. Brown and A. Romanov, “Contravariant forms on Whittaker modules,” *Proceedings of the American Mathematical Society*, vol. 149, no. 1. American Mathematical Society, pp. 37–52, 2021.","mla":"Brown, Adam, and Anna Romanov. “Contravariant Forms on Whittaker Modules.” *Proceedings of the American Mathematical Society*, vol. 149, no. 1, American Mathematical Society, 2021, pp. 37–52, doi:10.1090/proc/15205.","ama":"Brown A, Romanov A. Contravariant forms on Whittaker modules. *Proceedings of the American Mathematical Society*. 2021;149(1):37-52. doi:10.1090/proc/15205","ista":"Brown A, Romanov A. 2021. Contravariant forms on Whittaker modules. Proceedings of the American Mathematical Society. 149(1), 37–52.","apa":"Brown, A., & Romanov, A. (2021). Contravariant forms on Whittaker modules. *Proceedings of the American Mathematical Society*. American Mathematical Society. https://doi.org/10.1090/proc/15205"},"date_created":"2020-11-19T10:17:40Z","department":[{"_id":"HeEd"}],"type":"journal_article","quality_controlled":"1","language":[{"iso":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication":"Proceedings of the American Mathematical Society","ec_funded":1,"issue":"1"},{"scopus_import":1,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1810.07039"}],"date_published":"2019-11-01T00:00:00Z","oa_version":"Preprint","doi":"10.1090/proc/14586","oa":1,"ec_funded":1,"issue":"11","publication":"Proceedings of the American Mathematical Society","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","type":"journal_article","language":[{"iso":"eng"}],"department":[{"_id":"TaHa"}],"date_created":"2019-11-04T16:10:50Z","citation":{"chicago":"Li, Penghui. “A Colimit of Traces of Reflection Groups.” *Proceedings of the American Mathematical Society*. AMS, 2019. https://doi.org/10.1090/proc/14586.","short":"P. Li, Proceedings of the American Mathematical Society 147 (2019) 4597–4604.","ieee":"P. Li, “A colimit of traces of reflection groups,” *Proceedings of the American Mathematical Society*, vol. 147, no. 11. AMS, pp. 4597–4604, 2019.","mla":"Li, Penghui. “A Colimit of Traces of Reflection Groups.” *Proceedings of the American Mathematical Society*, vol. 147, no. 11, AMS, 2019, pp. 4597–604, doi:10.1090/proc/14586.","ama":"Li P. A colimit of traces of reflection groups. *Proceedings of the American Mathematical Society*. 2019;147(11):4597-4604. doi:10.1090/proc/14586","apa":"Li, P. (2019). A colimit of traces of reflection groups. *Proceedings of the American Mathematical Society*. AMS. https://doi.org/10.1090/proc/14586","ista":"Li P. 2019. A colimit of traces of reflection groups. Proceedings of the American Mathematical Society. 147(11), 4597–4604."},"publisher":"AMS","title":"A colimit of traces of reflection groups","page":"4597-4604","publication_status":"published","day":"01","author":[{"last_name":"Li","first_name":"Penghui","id":"42A24CCC-F248-11E8-B48F-1D18A9856A87","full_name":"Li, Penghui"}],"external_id":{"arxiv":["1810.07039"]},"date_updated":"2021-01-12T08:11:20Z","publication_identifier":{"issn":["0002-9939","1088-6826"]},"year":"2019","article_type":"original","volume":147,"intvolume":" 147","status":"public","_id":"6986","project":[{"name":"Arithmetic and physics of Higgs moduli spaces","call_identifier":"FP7","grant_number":"320593","_id":"25E549F4-B435-11E9-9278-68D0E5697425"}],"abstract":[{"text":"Li-Nadler proposed a conjecture about traces of Hecke categories, which implies the semistable part of the Betti geometric Langlands conjecture of Ben-Zvi-Nadler in genus 1. We prove a Weyl group analogue of this conjecture. Our theorem holds in the natural generality of reflection groups in Euclidean or hyperbolic space. As a corollary, we give an expression of the centralizer of a finite order element in a reflection group using homotopy theory. ","lang":"eng"}],"month":"11"},{"publication_identifier":{"issn":["0002-9939","1088-6826"]},"extern":"1","date_published":"2015-12-21T00:00:00Z","article_processing_charge":"No","doi":"10.1090/proc/12796","oa_version":"None","volume":144,"year":"2015","article_type":"letter_note","title":"A note on micro-instability for Hamiltonian systems close to integrable","publication_status":"published","day":"21","page":"1553-1560","date_updated":"2021-01-12T08:19:40Z","author":[{"first_name":"Abed","last_name":"Bounemoura","full_name":"Bounemoura, Abed"},{"full_name":"Kaloshin, Vadim","orcid":"0000-0002-6051-2628","last_name":"Kaloshin","first_name":"Vadim","id":"FE553552-CDE8-11E9-B324-C0EBE5697425"}],"date_created":"2020-09-18T10:46:14Z","_id":"8495","citation":{"short":"A. Bounemoura, V. Kaloshin, Proceedings of the American Mathematical Society 144 (2015) 1553–1560.","chicago":"Bounemoura, Abed, and Vadim Kaloshin. “A Note on Micro-Instability for Hamiltonian Systems Close to Integrable.” *Proceedings of the American Mathematical Society*. American Mathematical Society, 2015. https://doi.org/10.1090/proc/12796.","mla":"Bounemoura, Abed, and Vadim Kaloshin. “A Note on Micro-Instability for Hamiltonian Systems Close to Integrable.” *Proceedings of the American Mathematical Society*, vol. 144, no. 4, American Mathematical Society, 2015, pp. 1553–60, doi:10.1090/proc/12796.","ieee":"A. Bounemoura and V. Kaloshin, “A note on micro-instability for Hamiltonian systems close to integrable,” *Proceedings of the American Mathematical Society*, vol. 144, no. 4. American Mathematical Society, pp. 1553–1560, 2015.","ama":"Bounemoura A, Kaloshin V. A note on micro-instability for Hamiltonian systems close to integrable. *Proceedings of the American Mathematical Society*. 2015;144(4):1553-1560. doi:10.1090/proc/12796","apa":"Bounemoura, A., & Kaloshin, V. (2015). A note on micro-instability for Hamiltonian systems close to integrable. *Proceedings of the American Mathematical Society*. American Mathematical Society. https://doi.org/10.1090/proc/12796","ista":"Bounemoura A, Kaloshin V. 2015. A note on micro-instability for Hamiltonian systems close to integrable. Proceedings of the American Mathematical Society. 144(4), 1553–1560."},"abstract":[{"text":"In this note, we consider the dynamics associated to a perturbation of an integrable Hamiltonian system in action-angle coordinates in any number of degrees of freedom and we prove the following result of ``micro-diffusion'': under generic assumptions on $ h$ and $ f$, there exists an orbit of the system for which the drift of its action variables is at least of order $ \\sqrt {\\varepsilon }$, after a time of order $ \\sqrt {\\varepsilon }^{-1}$. The assumptions, which are essentially minimal, are that there exists a resonant point for $ h$ and that the corresponding averaged perturbation is non-constant. The conclusions, although very weak when compared to usual instability phenomena, are also essentially optimal within this setting.","lang":"eng"}],"month":"12","publisher":"American Mathematical Society","issue":"4","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication":"Proceedings of the American Mathematical Society","intvolume":" 144","language":[{"iso":"eng"}],"status":"public","quality_controlled":"1","type":"journal_article"}]