# What happened to the fourth paper in the series “On the classification of primitive ideals for complex classical Lie algebras” by Garfinkle?

In a series of papers in Compositio Math. entitled On the classification of primitive ideals for complex classical Lie algebras I, II and III, Garfinkle describes an algorithm that allows one to determine the fibers of the Duflo map, and hence the left (or right) Kazhdan-Lusztig cells in the Weyl gorup for a Lie algebra of type $B$ or $C$.

Several places, both in reviews and in the papers themselves, mention is made of a fourth part which will deal with the additional things needed to treat type $D$, but I have not been able to find this paper anywhere (it is certainly not listed on MathSciNet).

What happened to this paper? Was it ever written? And if not, is there some other paper that describes what happens in type $D$?

• There was a question, since closed, on "Never appeared forthcoming papers," mathoverflow.net/questions/48477/… – Gerry Myerson May 10 '17 at 23:09
• @GerryMyerson, since this paper does not seem to appear as an answer there, do you suggest that it should have been posed as an answer (which I guess is now impossible, since your linked question is closed)? – LSpice May 11 '17 at 17:26
• @LSpice, this paper would have been a good addition to the answers to that question but, as you note, that was impossible. That question was closed before Tobias posted this one. You may have noticed that I posted a comment on that question, linking to this one. – Gerry Myerson May 11 '17 at 23:00

It was written, but never published.

Tyson Gern's 2013 thesis references it:

• D. Garfinkle. On the classification of primitive ideals for complex classical Lie algebras, IV. unpublished.

Fortunately, the same thesis discusses the proof of the type $D$ case (page 41), and gives some extra references, including another article by Garfinkle, which are likely to be relevant.

You also might want to consider contacting the author of the thesis.

• Thank you, this is precisely the sort of thing I was looking for. I wish I had known about that thesis before reading the original papers, as the original explanation is nowhere near as enlightening as the one in the thesis (which is also essentially the version I had arrived at after spending many hours trying to understand the algorithms involved). – Tobias Kildetoft Nov 10 '15 at 8:42
• By the way, Devra Garfinkle got her Ph.D. at MIT (working with Michele Vergne and influenced by David Vogan) in 1982 but didn't remain active in mathematics. The series of three papers mentioned here can be accessed online at numdam.org, and were her last formal publications. – Jim Humphreys Nov 10 '15 at 14:45
• @JimHumphreys Thank you. It is always nice to hear a bit more about the background in cases like this. – Tobias Kildetoft Nov 15 '15 at 21:22

I am the author of this series of papers. Thank you for your interest in my work. I have continued to be active in math when I have had time for it, but most of my time in the past years has been taken with parenting responsibilities.

After completing the third paper, I began work on the fourth paper and made partial progress before setting it aside for an number of years, though I had hoped to return to it.

Because of interest in this work by researchers in this area, I had shared an incomplete draft of the fourth paper with some colleagues at their request. Last year, to my surprise, I discovered that a link to a modified version of that draft had been posted to this website. At my request, it has been taken down. To be totally clear, this version did not and does not have my authorization to have my name on it, nor to use my copyrighted content.

Meanwhile, I have been working on completing the fourth paper. While much of the proof parallels the argument in the third paper, there are new difficulties that need to be resolved. Many of them have to do with the fact that the Weyl group for type D_n is of index 2 in the Weyl group of type C_n. Thus, you have fewer pairs of tableaux to work with, and the proofs are a lot more difficult. However, as a result of recent progress, I am now optimistic that in the near future I will have a manuscript that can be posted to the arXiv.

• Welcome to the site! When you say "a modified version … had been posted to this website", you surely don't mean MathOverflow by "this site", right? As far as I know, MO doesn't have any facility for hosting papers. – LSpice May 10 '17 at 16:43
• I'm sorry. My bad. I meant to say that a link had been posted. I'll fix it. – Devra Garfinkle Johnson May 10 '17 at 17:52
• I'm mostly curious, but -- is there an implied subtext that the bootleg version of the 4th paper that appeared here earlier has incorrect proofs? (Kudos on returning to unfinished business, by the way. Saying this as someone who usually tires out after type A...) Also, welcome to MO! – darij grinberg May 10 '17 at 19:42
• Writing this paper is a chore, or worse. If I thought the bootleg version had good proofs, I would happily put my name on it and save myself the trouble. – Devra Garfinkle Johnson May 10 '17 at 22:15

Gu's UROP+ paper "Nilpotent orbits: Geometry and combinatorics" references a recent version of the fourth paper of the series. Unfortunately, the referenced link is broken, so I'm not sure where the paper is accessible. You may want to contact William McGovern for a copy.