What type of letter is Pierre Deligne talking about and why did he write a letter to other mathematicians?
I just did something like this last month...
Well, with modern technology available, I didn't write a letter; I wrote up some notes, scanned them in, and attached them to an e-mail.  Keep in mind, though, that Prof. Deligne started his career in the days before e-mail, and telephone calls (from rotary dial phones attached to landlines) were expensive (like a dollar a minute.)
I had an alternate approach to proving a theorem in a paper I had read and thought perhaps the approach could also generalize beyond their theorem.  I wrote up a sketch of the alternate proof (for which some details I have not actually proved) in 3 pages of notes.  I scanned them, and attached them to an email to the authors that read something like &quot;Dear X and Y, I hope you are well.  We talked about this at Online Conference Z and I'm attaching some notes.  I'm happy to meet by Zoom and talk.  Best, -me&quot;
I didn't write up a paper because it would take 10 pages and at least a couple weeks of work to give precise definitions of everything, and any paper would only be publishable in a fairly low-tier journal (since the result isn't new and the approach also isn't new, having been applied to other problems before), which at this point in my career doesn't count for anything.