Draft:Gustav Isaak Lehrer

Gus Lehrer (born in Munich, Germany, 18th January, 1947) is an Australian German- born mathematician, and Emeritus Professor at the University of Sydney. He was President of the Sydney Jewish Museum for 11 years, from 2010 to 2021.

Education and Career

Born in Munich to Polish-Jewish Holocaust survivors Leon Lehrer and Eugenie Weissager, he emigrated from Germany to Australia with his parents in 1950, on the refugee ship Cyrenia. He did his undergraduate degree at Sydney University, graduating with the University Medal and Barker Prize in Pure Mathematics in 1967. In 1968 he received a Commonwealth Scolarship to study for his PhD in England. He went to the University of Warwick to do his PhD, under the supervision of J.A. (James Alexander, better known as “Sandy”) Green.

After completing his PhD on the representation theory of the finite special linear groups in 1971, Lehrer remained in the UK for 3 years, accepting lectureships at the University of Warwick, then Birmingham. In 1974, he went to Sydney University to take up a lectureship. He has remained there for the rest of his career. After a brief period as Reader, he was promoted to a personal Chair in 1988. More details may be found in the interview [5] conducted by Peter Hall.

During 1996–1998, Lehrer accepted the position of Director of the Centre for Mathematics and its Applications (CMA) at the Australian National University.

Research Contributions

Lehrer is known for his contributions in four areas:

• the representation theory of reductive groups over finite fields
• configuration spaces
• reflection groups
• invariant theory.

Highlights from these four areas are as follows.

In reductive groups, he proved (wth R. Howlett) [2] a “decomposition theorem” for induced cuspidal representations, confirming a conjecture of T.A. Springer. This theory, (now known as “Howlett-Lehrer theory”) has very wide applications. The second highlight in this area is Lehrer’s definition and study [1] of Cellular Algebras (with J. Graham), instrumental in deformation theory. In configuration spaces, Lehrer developed three methods for computing the action of an automorphism on the cohomology of a hyperplane complement [3]: differential, ℓ-adic and Z-function”. He wrote (with D. Taylor) the definitive work on unitary reflection groups [7] and in invariant theory solved the second fundamental theorem for classical groups (with R. Zhang) [4] and pioneered the use of categorical diagrammatic methods [6].

Awards and Honours

• Kenna Distinguished Lecturer, Notre Dame (U.S.A.), 1990.
• Adjunct Professor, Australian National University since 1998.
• Fellow of the Australian Academy of Science since 1997.
• Alexander von Humboldt Research Prize, 1999.
• Colloque Solstice d'été, Paris, 2003, 2009.
• Australian Centenary Medal 2003.
• Special Issue in honour of GL (Vol 311), Journal of Algebra, 2009.
• Ordway Distinguished Lectures, University of Minnesota, 2010.
• Hannan Medal of the Australian Academy of Science, 2015.
• George Szekeres Medal of the Australian Mathematical Society, 2016.
• Member of the Order of Australia (AM), 2015.
• BICMR Peking University Distinguished Lecture Series, 2019.
• Aisenstadt Chair, Montreal, 2022.

References

[1] G.I. Lehrer (with J.Graham), “Cellular algebras”, Inventiones Math. 123 (1996), 1–34.
[2] G.I. Lehrer and R. Howlett, “Induced cuspidal representations and generalised Hecke rings”, Inventiones Math. 58, 37-64 (1980)
[3] G.I. Lehrer, "Poincaré series for unitary reflection groups", Inventiones Math. 120 (1995), 411-425.
[4] G.I. Lehrer and R.B. Zhang, "The second fundamental theorem of invariant theory for the orthogonal group", Annals of Mathematics 176 (2012), 2031-2054.
[5] Hall, Peter, "Interview with Gus Lehrer" Asia Pac. Math. Newsl. 2 (2012), no. 2, 30-36.
[6] Lehrer, G. I. and Zhang, R. B., "The Brauer category and invariant theory", J. Eur. Math. Soc. (JEMS) 17 (2015), no. 9, 2311-2351.
[7] Lehrer, Gustav I. and Taylor, Donald E., "Unitary reflection groups", Austral. Math. Soc. Lect. Ser., 20, Cambridge University Press, Cambridge, 2009, viii+294 pp. ISBN: 978-0-521-74989-3