# Symplectic Steinberg group

I have several questions about Steinberg group and K2 for symplectic group:

1. Can I extend the definition of Steinberg symbols to symplectic case? Will they generate the center of Steinberg group?
2. Does the center of symplectic Steinberg group coincide with K2 (the kernel of $\mathrm{SpSt}\rightarrow\mathrm{Sp}$ as usual)?
3. Is there an analogue for Matsumoto's theorem?

I tryed to read "Sur les sous-groupes arithmetiques des groupes semi-simples deployes" by Hideya Matsumoto and all I got to know about symplectic case is that there is some problems with long roots in Cl. Also, is it written in english anywhere about non-Al K-theory? There is "The Classical groups and K-theory" by Hahn and O'Meara, but it tells about SL and about unitary groups only.

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Your questions cover a lot of territory, so there is by now a lot of literature including extensive work on the congruence subgroup problem and on algebraic K-theory. It might help to separate the questions and give a little more context. – Jim Humphreys Apr 2 '10 at 21:23
In fact the unitary groups of Hahn and O'Meara cover the symplectic case: See their example (A) in 5.2B. – Wilberd van der Kallen Oct 3 '11 at 10:05

There is useful information about the symplectic analogues of the Steinberg group, the Steinberg symbols, and the $K_2$ functor in some of Michael Stein's papers from the '70's. In particular, see his papers "Generators, relations and coverings of Chevalley groups over commutative rings" and "Surjective stability in dimension $0$ for $K_{2}$ and related functors" and "Injective stability for $K_{2}$ of local rings".

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Hyman Bass and his students (including Anthony Bak, Michael Stein) in the early 1970s did a lot of work on $K_2$ involving classical groups besides those of type $A$, before Quillen's success with the higher $K$-groups. The typewritten lecture notes by Bass Algebraic K-Theory (W.A. Benjamin, 1968) contain good foundational material but are now hard to find and naturally aren't up to date. – Jim Humphreys Apr 3 '10 at 12:26
There exists a pretty good scan in djvu format of Bass's book floating around the internet. Google should be able to locate it... – Andy Putman Apr 3 '10 at 15:36

Here's a small follow-up on Matsumoto's thesis, which deals essentially with the congruence subgroup problem for Chevalley (split) algebraic groups. This followed work by Bass-Milnor-Serre, but in turn was followed by more technical work on nonsplit groups (Prasad-Raghunathan, in particular). In my 1980 Springer Lecture Notes 789 on Arithmetic Groups, I tried in the last part to convey Matsumoto's ideas in the special case of $SL_n$ when $n \geq 3$ but with some side remarks about the general case. The congruence subgroup problem has a different solution for $n=2$, which should for this and some other purposes be assigned to type $C_1$ rather than the conventional $A_1$. For symplectic groups there is a significant difference, mentioned in my Remark on page 129. Briefly:

The ideas in my $SL_n$ proof carry over almost unchanged to other Chevalley groups of rank $\geq 2$, with one important modification due to the fact that real Lie groups of type $C$ starting with rank 1 have an infinite cyclic fundamental group while others have a finite fundamental group. This is what complicates the proof for symplectic groups in Matsumoto's paper (which is only available in French). The special feature of root systems of type $C$ seems to be the existence of roots equal to twice a weight.

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There is an excellent survey paper: Linear Algebraic Groups and K-theory http://users.ictp.it/~pub_off/lectures/lns023/Rehmann/Rehmann.pdf by Ulf Rehmann. It seems that Matsumoto paper concerns symplectic case, that is the answers to all your questions are positive.

Note that for non-symplectic groups, Steinberg symbols are bilinear, however for symplectic it is not true. For a nice description of the 2-cocycle of the topological universal cover $\widetilde{SL_2({\mathbb R})}$ of $SL_2({\mathbb R})$ see e.g. Asai, T.: The reciprocity of Dedekind sums and the factor set for the universal covering group of $SL(2,{\mathbb R})$.

From Asai work, one can deduce that the Steinberg Symbol corresponding to a $\widetilde{SL_2({\mathbb R})}$ is defined as: for $x,y\in {\mathbb R}^{\times}$

$c(x,y) = \left\{ \begin{array}{l l} -1 & \quad \text{if } x < 0 \text{ and } y<0 \\ 0 & \quad \text{otherwise}\\ \end{array} \right.$

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Let me do a bit of necromancy here and address the third question.

A Note on Milnor–Witt K-Theory and a Theorem of Suslin by K. Hutchinson and L. Tao provides a description for $H_2\left(Sp(F),\mathbb{Z}\right)=H_2\left(SL(2,F),\mathbb{Z}\right)$ for an infinite field $F$ as Milnor—Witt K-theory $K_2^{MW}(F)$, introduced by F. Morel in 2003 in his study of $\mathbb{A}^1$-homotopy theory.

$K_*^{MW}(F)$ is a graded associative ring generated by the symbols $[u]$, $u\in F^*$ of degree $+1$ and one symbol $\eta$ of degree $-1$ modulo the following relations:

• For $a\in F\setminus\{0,1\}$, $[a]\cdot[1-a]=0$;
• For $a,b,\in F^*$, $[ab]=[a]+[b]+\eta[a][b]$;
• For $u\in F^*$, $[u]\eta=\eta[u]$;
• $\eta^2[-1]+2\eta=0$.

The proof is based on Matsumoto—Moore presentation for $H_2\left(Sp(F),\mathbb{Z}\right)$ and the coincidence of $K_2^{MW}(F)$ with $K_2^{MM}(F)$.

PS. The equality $H_2\left(Sp(F),\mathbb{Z}\right)=H_2\left(SL(2,F),\mathbb{Z}\right)$ has something to do with $\mathsf{A}_1=\mathsf{C}_1$ (see this MO question).

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