URL needs to be in certain case letters

You can try something like this:

when HTTP_REQUEST {
    if { [HTTP::uri] equals "WebCenter" } {
         URL is correct case, do nothing"
    }
    elseif { string tolower [HTTP::uri] equals "webcenter" }
        HTTP::redirect "https://qweb.sys.com/WebCenter/default with a capital W and a capital C"
    }
}

Or, do you want to just fix it for them?

You can try something like this:

when HTTP_REQUEST {
    if { [HTTP::uri] equals "WebCenter" } {
         URL is correct case, do nothing"
    }
    elseif { string tolower [HTTP::uri] equals "webcenter" }
        HTTP::redirect "https://qweb.sys.com/WebCenter/default with a capital W and a capital C"
    }
}

Or, do you want to just fix it for them?

I should have read your question a little close before I responded the first time. Here is an example where the user is redirected with the proper case for WebCenter and includes the rest of the URL. I haven't actually tested this, so there could be some syntax issues.

when HTTP_REQUEST {
    if { [HTTP::uri] starts_with "WebCenter" } {
         URL is correct case, do nothing"
    }
    elseif { string tolower [HTTP::uri] starts_with "webcenter" }
         redirect user to the proper case, including the remainter of the URI.
         The value 10 in the substring command should be the first character of the rest of the URI.
        HTTP::redirect "https://qweb.sys.com/WebCenter/[substr [HTTP::uri] 10]"
    }
}

Please try this. Hope it will work as per your plan.

    when HTTP_REQUEST { 
          if { ( [string tolower [HTTP::uri]] starts_with "/webcenter" ) } { 
                HTTP::redirect https://qweb.sys.com/WebCenter/default
              } 
            }

Am I correct to assume that the origin web server needs to receive the request in the proper case, but that it doesn't really matter from a client-perspective?

If so, I would recommend re-writing the URI regardless of the case of the request. Redirecting does the trick, but it requires another request (i.e. increased overhead) and isn't as clean as a re-rewrite in my opinion. Something like this should do the trick if you want to re-write the request transparently:

when HTTP_REQUEST {

    if { [string tolower [HTTP::path]] starts_with "/webcenter/" } {

        HTTP::uri "[string map -nocase { "/webcenter/" "/WebCenter/" } [HTTP::path]]"

    }

}

(Note - I haven't tested the above code so I hope I didn't make any typos/omissions.)

this is just one of examples.

 configuration

root@(ve11c)(cfg-sync In Sync)(Active)(/Common)(tmos) list ltm virtual bar
ltm virtual bar {
    destination 172.28.24.10:80
    ip-protocol tcp
    mask 255.255.255.255
    pool foo
    profiles {
        http { }
        tcp { }
    }
    rules {
        qux
    }
    source 0.0.0.0/0
    source-address-translation {
        type automap
    }
    vs-index 4
}
root@(ve11c)(cfg-sync In Sync)(Active)(/Common)(tmos) list ltm rule qux
ltm rule qux {
    when HTTP_REQUEST {
  switch -glob [HTTP::uri] {
    "/[wW]ebcenter/default" -
    "/WEBCENTER/default" {
       Do nothing
    }
    "/[wW][eE][bB][cC][eE][nN][tT][eE][rR]/default" {
      HTTP::redirect "http://[HTTP::host]/WebCenter/default"
    }
    default {
       Do nothing
    }
  }
}
}

 test

[root@ve11c:Active:In Sync] config  curl -I http://qweb.sys.com/webcenter/default
HTTP/1.1 404 Not Found
Date: Sun, 24 May 2015 14:18:07 GMT
Server: Apache/2.2.3 (CentOS)
Last-Modified: Sun, 09 Feb 2014 08:39:51 GMT
ETag: "41879c-59-2a9c23c0"
Accept-Ranges: bytes
Content-Length: 89
Content-Type: text/html; charset=UTF-8

[root@ve11c:Active:In Sync] config  curl -I http://qweb.sys.com/Webcenter/default
HTTP/1.1 404 Not Found
Date: Sun, 24 May 2015 14:18:13 GMT
Server: Apache/2.2.3 (CentOS)
Last-Modified: Sun, 09 Feb 2014 08:39:51 GMT
ETag: "41879c-59-2a9c23c0"
Accept-Ranges: bytes
Content-Length: 89
Content-Type: text/html; charset=UTF-8

[root@ve11c:Active:In Sync] config  curl -I http://qweb.sys.com/WEBCENTER/default
HTTP/1.1 404 Not Found
Date: Sun, 24 May 2015 14:18:32 GMT
Server: Apache/2.2.3 (CentOS)
Last-Modified: Sun, 09 Feb 2014 08:39:51 GMT
ETag: "41879c-59-2a9c23c0"
Accept-Ranges: bytes
Content-Length: 89
Content-Type: text/html; charset=UTF-8

[root@ve11c:Active:In Sync] config  curl -I http://qweb.sys.com/WeBcEnTeR/default
HTTP/1.0 302 Found
Location: http://qweb.sys.com/WebCenter/default
Server: BigIP
Connection: Keep-Alive
Content-Length: 0